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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 1 Data-driven motion compensation techniques for noncooperative ISAR imaging Risto Vehmas, Student Member, IEEE, Juha Jylhä, Minna Väilä, Juho Vihonen, Member, IEEE, and Ari Visa, Senior Member, IEEE Abstract—We consider the data-driven motion compensation problem in inverse synthetic aperture radar (ISAR) imaging. We present optimization-based ISAR techniques and propose improvements to the range alignment, time-window selection, autofocus, time-frequency-based image reconstruction and crossrange scaling procedures. In experiments, the improvements reduced the computational burden and increased the image contrast by 50 percent at best and 28 percent on average in several test cases including changing translational and rotational motion. Index Terms—Inverse synthetic aperture radar, radar imaging, motion compensation, optimization, image contrast I. I NTRODUCTION Inverse synthetic aperture radar (ISAR) is a well-established technique for high-resolution radar imaging of moving objects. The principles of ISAR are analogous to synthetic aperture radar (SAR) and have been introduced in the literature in great detail in the 1980s (e.g., in [1]–[3]). A two-dimensional highresolution image of an object provides beneficial information for target recognition applications. This imaging capability renders ISAR a valuable tool in applications such as maritime and air surveillance. The traditional arrangement of noncooperative ISAR is to consider a stationary monostatic radar and a moving object. The radar obtains a time series of high-resolution range profiles using a suitable high-bandwidth waveform. The large antenna required for high cross-range resolution is synthesized by exploiting the relative motion between the radar and the object. The synthetic phased array antenna created by the object in different spatial positions can in principle be steered and focused to produce high cross-range resolution. However, to be able to focus this synthetic phased array antenna correctly, the relative motion has to be known very accurately. When the object is noncooperative, this motion is unknown and has to be estimated. This situation calls for what is referred to as data-driven motion compensation. ISAR techniques can also be used to image moving objects using a space- or airborne SAR sensor if the relative motion between the radar and the object is suitable. This subject was first introduced in Raney’s paper [4] and has been studied in the SAR literature ever since, attracting a great deal of interest more recently [5]–[12]. These techniques have an increasing number of applications in space-, air-, and ground-based radar Authors’ address: Tampere University of Technology, Laboratory of Signal Processing, P.O. Box 553, FI-33101, Tampere, Finland, e-mail: ([email protected]) imaging. The trend of this technology is moving towards increased spatial resolution, which motivates the development of new algorithms that are capable of delivering enhanced performance, preferably in real time. A. Related work on data-driven motion compensation Traditionally, the problem of data-driven motion compensation in ISAR is divided into several consecutive steps. The reason for this is simple: the problem is too complex and time-consuming to solve using a more direct approach. In a two-dimensional setting, a direct non-approximated approach would require three degrees of freedom to be estimated for each element of the synthetic phased array antenna (the position and orientation of the object). To make the problem tractable, suitable approximations and simplifying choices are needed in the development of an ISAR imaging algorithm. In ISAR processing, a common practice is to decompose the motion of the object into two parts: translational motion and rotational motion. These concepts are defined more carefully in Section II, but the important distinction between them is their significance for cross-range processing. The translational motion is the undesired component, whereas the rotational motion is essential and required for obtaining cross-range resolution. The translational motion needs to be taken into account in the imaging process, and its compensation is usually the first step of the ISAR algorithm. Translational motion compensation is usually performed in two parts: range alignment and autofocus. Range alignment is used to compensate for translational motions larger in magnitude than the range resolution [13]. A widely used approach is to minimize (or maximize) a quality measure calculated from the shifted amplitude envelopes of the range profiles [14]–[19]. After the range alignment procedure, the residual translational motions are corrected using an autofocus algorithm. Autofocus algorithms have been studied extensively in the literature; the most prominent techniques are based either on the phase gradient autofocus algorithm [20], [21] or contrast optimization autofocus [14], [22]–[25]. After the translational motion is compensated for, the Doppler histories caused by the rotational motion can be used to obtain a high cross-range resolution. To obtain a correctly focused image, the nature of the rotational motion needs to be taken into account in the image reconstruction. The process of estimating the unknown rotational motion and using it in the image reconstruction is generally more complicated than the translational motion compensation. The simplest way to perform the image reconstruction is with a one-dimensional 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 2 Fourier transform, that is, to assume a linear phase progression across the synthetic aperture. However, when the range resolution is extremely high and a similarly high resolution is desired in the cross-range direction, this simple range-Doppler algorithm rarely produces an acceptable result. The reason for this is the following: as the rotational angle increases, scatterers start to migrate through range resolution cells and the linear phase approximation breaks down. Moreover, if the rotational motion is not uniform, the phase progression is nonlinear even during a short time interval. As solutions to these problems, a technique called keystone formatting partially compensates for the range cell migration [6], [9], [26], [27], while time-frequency representations are used to mitigate the effects of nonlinear phase histories [28]–[33]. The former approach was first introduced by Perry et al. [6] in the context of airborne SAR imaging of moving objects. The approach based on time-frequency representations was introduced by Chen [28], [30] in the 1990s and has since matured into a well-established technique [29]–[38] in ISAR processing. For example, the S-method has been demonstrated to greatly surpass the conventional Fourier transform-based range-Doppler approach in [37]. Performing the ISAR image reconstruction based on the aforementioned techniques eliminates the need for explicitly estimating the rotational motion parameters of the object. These parameters are, however, required to determine the spatial cross-range scale of the resulting ISAR image. This problem, which is called cross-range scaling in the ISAR literature, is generally quite difficult to solve. Existing approaches are based on either estimating high-order phase coefficients from the range-compressed signal [39]–[41] or a suitable optimization approach [42], [43]. The optimization approaches rely on dividing the synthetic aperture into two or more parts to produce multiple ISAR images, which are scaled and rotated versions of each other. B. Novel contributions detailed Producing a very high cross-range resolution (≈ 10 cm) in the presence of complicated target motions is an open problem within the ISAR imaging community. Previously, several approaches have been shown to produce satisfying results in the individual parts of the motion compensation [15], [17], [19], [22], [25], [42], [43], but often at the cost of a high computational burden. However, the aforementioned papers and references therein define several suitable approaches for different parts of the data-driven motion compensation process. We use and combine them into a computationally efficient noncooperative ISAR imaging algorithm based on mathematical optimization. Our novel contributions can be listed as follows: 1) for the global range alignment method [15], we propose using a carefully selected initial guess, after which we derive an expression for the gradient of the loss function and propose new loss functions. Together, these novelties lead to increased estimation performance and reduced computational burden; 2) we propose including the keystone formatting and autofocus procedures in the well-established time window optimization procedure [44], yielding a higher cross-range resolution in cases with randomness in target motion over a coherent processing interval; 3) we show that an expression for the second order partial derivatives of the loss function can be derived, which improves the contrast optimization autofocusing when supplemented with low-complexity mathematical optimization; 4) we show how the rotation correlation [42] and polar mapping methods [43] can be efficiently combined to provide a straightforward solution to the cross-range scaling problem; and 5) we propose using a contrast optimization procedure for the time-frequency representations typical in ISAR imaging. This procedure reduces the spatially variant blurring caused by the nonuniform rotational motion of the target. Together, these contributions yield a higher spatial resolution under complex target motion dynamics, which is experimentally validated using an X-band ISAR system. In reference to the well-established optimization-based ISAR processing, the performance is improved by some 28 percent as measured by the image contrast, a standard quality measure in the cited literature, with a lower computational cost. The paper is organized in the following manner. Section II recalls the conventional ISAR signal model for a twodimensional imaging geometry. Then Section III presents the data-driven motion compensation and imaging algorithms as well as the proposed improvements for them which lead to our ISAR algorithm. Section IV demonstrates our ISAR algorithm with a numerical example using measured radar data. Finally, Section V draws conclusions. II. ISAR SIGNAL MODEL This Section analyzes the conventional ISAR signal model [13], [14], [30], [34], [38], [39], [41], [42], [44] and its underlying assumptions. The model is appropriate for a twodimensional imaging geometry with a moving object and monostatic stationary radar. The model we use is extensively discussed in references [34], [38]. The signal model is reviewed for two reasons. First, it is essential for understanding the significance of different parts of any ISAR algorithm. Second, as the desired resolution of the ISAR image increases, it is important to carefully examine each simplifying approximation that is made in the analysis. Readers familiar with basic ISAR principles may skip this Section and continue to Section III. Fig. 1 illustrates a two-dimensional ISAR geometry. The object of interest is moving in the (x, y) coordinate system. The (x0 , y 0 ) coordinate system (object frame of reference) is rigidly attached to the object and is a shifted and rotated version of the (x, y) system (radar frame of reference). These definitions mean that the motion of the object is completely described by the location of the origin of the object frame of reference r0 (the line-of-sight vector) and the angle θ between any unit vector in the object frame and r0 . The origin of the (x0 , y 0 ) system is assumed to be located in the object’s center of mass. As the aspect angle θ changes, the range ||Rp (t)|| between each different point on the object and the 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 3 radar changes uniquely. Slow-time t is the variable which increases along the synthetic aperture and remains constant during the formation of a single range profile. Fig. 1 represents the imaging geometry for a fixed value of t, and as the object moves, the primed coordinate system shifts and rotates accordingly. Assuming that the radar uses a pulsed waveform to produce a series of high-resolution range profiles, we made the usual start-stop approximation, meaning that the object is assumed to be stationary between the transmission and reception of a single pulse. We assumed that the principle of superposition applies for the back-scattered echoes, in which case the point target response (PTR) can be used to completely describe the received signal. Assuming a linearly frequency modulated (LFM) waveform and pulse compression are used, the PTR ssp : R2 → C can be expressed in closed form. After range compression and quadrature demodulation of the PTR as a function of radial distance (range) r and slow-time t is 4π 2B (r − Rp (t)) e−i λ Rp (t) , (1) ssp (r, t) = sinc c where λ is the carrier wavelength, Rp (t) = kRp (t)k, B is the temporal frequency bandwidth of the LFM waveform, and c is the propagation speed of the radio wave. Equation (1) can be derived by using the correlation theorem and the principle of stationary phase and assuming that the transmitted signal is band-limited [45]. The most important quantity in this analysis is Rp (t) since it determines both the range location of the amplitude envelope and the phase of the PTR (1). Fig. 1 has Rp = r0 − rp , from which Rp2 = r02 + rp2 − 2r0 · rp π = r02 + rp2 − 2r0 rp cos θp + θ + 2 = r02 + rp2 + 2r0 rp sin (θp + θ) (2) can easily be deduced. In (2) we have denoted r0 = kr0 k and rp = krp k. The aspect angle θ is defined as (see Fig. 1) r̂0 ·ŷ 0 = cos θ + π2 , where r̂0 = r0 /r0 and ŷ 0 is a unit vector pointing in the y 0 -direction. The angle θp is the angle between rp and ŷ 0 , which is a constant because the (x0 , y 0 ) coordinate system is assumed to be rigidly attached to the object. Notably, when the object is in the far field, we have r0 rp . This observation means that we are motivated to consider Rp as a function of rp and retain only terms up to the first order in its Taylor expansion. We have ∂Rp r0 sin (θp + θ) + rp , =q ∂rp r02 + rp2 + 2r0 rp sin (θp + θ) (3) which leads to ∂Rp (0) rp + O(rp2 ) ∂rp = r0 + rp sin (θp + θ) + O(rp2 ). Rp (rp ) = Rp (0) + (4) The second order term in the expansion (4) gives us a criterion that can be used to evaluate the applicability of the farfield approximation. Namely, the magnitude of the second order term should be significantly smaller than half the carrier wavelength λ. This leads to the condition rp2 λ cos2 (θp + θ) . r0 2 (5) Expanding the sine term in equation (4) and denoting yp = rp cos θp and xp = rp sin θp leads to Rp ≈ r0 + xp cos θ + yp sin θ. (6) In equation (6) xp and yp are constants that are the initial coordinates of the point scatterer in the primed coordinate system. As the object moves, both r0 and θ change as a function of slow-time t, so accounting for the motion of the object results in the expression Rp (t) ≈ r0 (t) + xp cos θ(t) + yp sin θ(t). (7) Under this approximation, the motion of each point can thus be divided into a spatially invariant (the same for every point p) term r0 , which is called translational motion, and a spatially variant (depends on p) part containing the θ-dependence, which is called rotational motion. The Fourier transform of the PTR (1) with respect to r can be expressed in closed form as Ssp (kr , t) = Fr→kr {ssp (r, t)} kr e−i2(kr +kc )Rp (t) =Π (2πB)/c kr c e−i2(kr +kc )[r0 (t)+xp cos θ(t)+yp sin θ(t)] , ≈Π 2πB (8) πB where kc = 2π/λ is the carrier frequency, kr ∈ [− πB c , c ] is the baseband spatial frequency variable, and Π is the rectangle function. Next, we assume that the object has a continuous reflectivity function g, which does not depend on θ or kr . Using the principle of superposition and for simplicity neglecting the effect of the band-limited nature of the signal (i.e. the rectangle function in (8)) we get Z ∞Z ∞ Ss(kr , t) = g(xp , yp )Ssp (kr , t)dxp dyp −∞ −∞ Z ∞Z ∞ 0 0 = e−i2kr r0 (t) g(x, y)e−i2kr [x cos θ(t)+y sin θ(t)] dxdy −∞ =e −i2kr0 r0 (t) G(2kr0 −∞ cos θ(t), 2kr0 sin θ(t)), (9) for the range-compressed signal as a function of (range) spatial frequency and slow-time. In (9), G(kx , ky ) = Fx→kx Fy→ky {g(x, y)} is the two-dimensional Fourier transform of the reflectivity function g, and kr0 = kr + kc is the pass-band spatial frequency variable. Equation (9) is interpreted as follows: under the far field assumption (7), the measured signal in the (kr , t)-domain is actually a phasemodulated slice of the two-dimensional Fourier transform of the reflectivity function g. Taking into account the bandlimited nature of the signal in the kr variable, these slices actually span an annular sector in the (kx , ky )-domain. The Fourier transform relationship between the received signal and the desired reflectivity function g is extremely useful and has 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 4 been analyzed thoroughly in the context of spotlight mode SAR [46]–[48]. For noncooperative objects, both r0 and θ are unknown in (9). Their values have to be estimated in order to reconstruct the ideal ISAR image g. To be more precise, since the signal is band-limited, the ideal image g cannot be exactly reconstructed, but the ideal reconstruction yields a convolution between g and the PTR. From this point of view, r0 is a nuisance that does not provide any useful information (since it is the same for every point (xp , yp )) and needs to be compensated for. Otherwise, the result of inverse Fourier transforming (9) will not result in a well-focused ISAR image. The following Section presents methods to estimate or compensate for these unknown motion parameters and reconstruct a properly focused and scaled ISAR image. III. DATA - DRIVEN MOTION COMPENSATION A. Range alignment The sum envelope of the range-compressed signal is defined as M −1 X p(r, ∆r) = |ss(r + ∆rm , tm )|, (11) m=0 where | · | denotes the absolute value and ∆r = [∆r0 . . . ∆rM −1 ]T . Using (11), the sharpness of the energy distribution of the sum envelope can be quantified by using a loss function Z ∞ L1 (∆r) = ψ(p(r, ∆r))dr, (12) −∞ where ψ : R → R is a real valued function of a real variable. For the contrast loss function ψ(p) = −p2 and ψ(p) = −p ln p for the entropy loss function. The sum of the squared envelope differences can be expressed as Z ∞M −2 X L2 (∆r) = [|ss(r + ∆rm+1 , tm+1 )| (13) −∞ m=0 2 The first step in our ISAR algorithm is the well-established range alignment procedure. The purpose of this procedure is to coarsely estimate the translational motion r0 (t), which is the slow-time-dependent distance between the origin of the object frame of reference and the radar. However, the estimate has to be more accurate than the range resolution if a good result is expected in the subsequent ISAR processing. In the first part of this subsection, we introduce the essential background of range alignment. Then, we describe our novel modifications. The first important observation from equations (1) and (7) is that compensating for the translational motion r0 (t) essentially means shifting each range profile ss(r, tm ) in the range direction by r0 (tm ), where m = 0, . . . , M − 1 and M is the number of samples in the slow-time direction. The idea of the global range alignment method [15] is to estimate these shifts by minimizing a loss function, whose value is calculated from the shifted amplitude envelopes of ss. The value of the loss function has to quantitatively measure the quality of the translational motion compensation. The advantage of the global method is that the entire signal is used in the loss function, whereas most other methods only use quality measures based on two adjacent range profiles. Another important observation is that the main effect of the translational motion is to shift the amplitude envelope of the back-scattered signal, while the shape of the range profile is preserved during a short slow-time interval. Thus, when the translational motion is properly compensated for, adjacent range profiles should be nearly identical. Quantitative measures for describing this similarity are, for example, the contrast and entropy of the sum envelope and the sum of the squared envelope differences [15], [17], [18]. Since the range-compressed signal is sampled, shifting the range profiles requires an interpolation. The ideal band-limited interpolation is most efficiently and accurately done in the Fourier domain by utilizing the convolution theorem (or in this case the shift theorem, equivalently) as ikr ∆rm ss(r + ∆rm , tm ) = Fk−1 e Fr→kr {ss(r, tm )} . r →r (10) −|ss(r + ∆rm , tm )|] dr. The range alignment problem is solved by minimizing L1 (or L2 ). By using a simple point scatterer model for g, it can be easily deduced that these loss functions contain multiple local minima. Thus, in this form the problem requires numerical global optimization, which is computationally very inefficient and does not guarantee that the optimal solution is found. The minimum entropy method [14] deals with this problem by using only two adjacent profiles in the definition of the sum envelope (11) and in (12). This results in M − 2 onedimensional optimization problems, but this method has its drawbacks. Namely, the method still requires one-dimensional global optimization, and target scintillation (rapid variations in the magnitude of g as the aspect angle θ changes) and error accumulation cannot be handled with success. The global method [15], [16] avoids error accumulation and can handle target scintillation but requires global optimization in a multidimensional search space even with a parametric model for the range shifts ∆r. By using a suitable heuristic optimization algorithm and a parametric model, the global method can be fairly effective [19]. However, the heuristic optimization algorithms are a compromise that does not guarantee an optimal solution and are thus better suited for offline processing than real-time scenarios. To overcome these issues, we propose the following modic m for the range shifts is fications. First, an initial estimate ∆r obtained by tracking the movement of the center of mass of the energy distribution of the range profiles, namely R∞ r|ss(r, tm )|2 dr c m = R−∞ . (14) ∆r ∞ |ss(r, tm )|2 dr −∞ We also note that another efficient way for obtaining the initial estimate is to use the maximum correlation method [49]. This means that the initial estimate is obtained from c m = arg max {|ss(r, t0 )| ?r |ss(r, tm )|} , ∆r (15) r where ?r denotes one-dimensional cross-correlation with respect to the range variable r. The initial estimates (14) or 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 5 (15) are typically not accurate enough to be used for the range alignment (as ∆r) themselves. This inaccuracy is the case especially when the range resolution is extremely high. Nevertheless, these estimates are accurate enough to turn the global optimization problem into a local problem, especially when a parametric model is used for the range shifts. A parametric model means that ∆rm = J X aj fj (tm ), (16) j=1 where {fj }j=1,...,J is a set of basis functions, and aj are their coefficients. These can be chosen, for example, to be Legendre polynomials (or any other orthogonal basis set), in which case the least squares estimates b aj for the coefficients of the initial estimate are easily obtained from b aj = M −1 1 X c ∆rm fj (tm ), Cj m=0 importance of the results in (19)–(21) becomes evident when they are combined with the first modification we proposed. The combination produces a range alignment algorithm that is computationally highly efficient. The final modification we propose is to combine the loss functions L1 and L2 . The loss functions (12) relying on the sum envelope work ideally when the change in θ is very small. As the change in θ during the coherent processing interval increases, scatterers start to migrate though range resolution cells, and the sum envelope becomes smeared even if r0 is properly compensated for. On the other hand, the loss function (13) works even if the shape of the range profile changes due to the rotational motion, since only adjacent envelopes are subtracted from each other. However, this again means that error accumulation and scintillation are not handled satisfactorily. A way to utilize the benefits of both is to combine them as (17) where the time variable is scaled such that tm ∈ [−1, 1], and the normalization factor is sX Cj = fj2 (tk ). (18) k Solving the range alignment problem is now equivalent to minimizing the loss function with respect to the coefficients aj of the parametric model (16) using (17) as the initial guess. Our second modification is to use first order numerical optimization to locate the minimum of the loss function. Assuming ψ is continuously differentiable, the partial derivatives of L1 with respect to aj can be obtained by applying the chain rule. For (12) we get Z ∞ M −1 ∂L1 ∂ψ(p(r, ∆r)) X ∂p(r, ∆r) ∂∆rm = dr ∂aj ∂p ∂∆rm ∂aj −∞ m=0 (19) M −1 X ∂L1 = fj (tm ) . ∂∆rm m=0 A straightforward calculation gives Z ∞ ∂L1 ∂ψ(p(r, ∆r)) ∂|ss(r + ∆rm , tm )| = dr. (20) ∂∆rm ∂p ∂r −∞ In the derivation of (20), we assume that r is a continuous variable, in which case the derivative property of the Fourier transform can be utilized. In practice, the partial derivatives ∂|ss|/∂r are evaluated using finite difference approximations. For the sum of squared envelope differences, a derivation with similar arguments gives Z ∞ ∂L2 ∂|ss(r + ∆rm , tm )| =2 [2|ss(r + ∆rm , tm )| ∂∆rm ∂r −∞ −|ss(r + ∆rm+1 , tm+1 )| − |ss(r + ∆rm−1 , tm−1 )|] dr. (21) If L2 is used as the loss function instead of L1 , we get the parametric gradient by replacing L1 with L2 in the last row of (19) (the result in the first row is not the same, but the chain rule yields the result in the last row in any case). The LRA (∆r) = H(L1 (∆r), L2 (∆r)), (22) where H : R2 → R is a function that combines L1 and L2 into a single loss function. For example, if ψ(p) = −p2 , a suitable choice would be H(L1 , L2 ) = −L2 /L1 . Similarly, for the entropy ψ(p) = −p ln p, the choices H(L1 , L2 ) = L1 + L2 and H(L1 , L2 ) = L1 L2 are useful. This means that in the last row of the parametric gradient (19) the expression X ∂H ∂Lj ∂LRA = , ∂∆r ∂Lj ∂∆r j (23) needs to be evaluated. In (23), ∂LRA /∂∆r denotes the gradient vector whose mth component is ∂LRA /∂∆rm . The loss function (22) could also be a summation of different functions H, but even a single function H brings out the benefits of both L1 and L2 . In summary, our range alignment algorithm consists of the following steps. First, an initial estimate is obtained using (14) or (15), after which the least squares estimates for the parametric model of order J are obtained using (17). After this, the minimum of the loss function (22) is located by using a first order numerical optimization algorithm (for example the steepest descent, nonlinear conjugate gradient, or quasiNewton method) with a line search procedure. An iterative procedure ∆r k+1 = ∆r k − αg k (24) is used, where g k is a descent direction obtained by using the gradient ∂LRA /∂a, and a = [a1 . . . aJ ]T . In (24), the superscripts stand for the number of iteration, ∆r 0 is the initial estimate (obtained by using either (14) or (15)), and the step length α is obtained by minimizing the function η k (α) = LRA (∆r k − αg k ) using a line search procedure. Once the estimate ∆r ? that minimizes the loss function LRA is obtained, it is used to compensate for the translational motion r0 . Switching back to continuous notation ∆r ? → rb0 (t), the coarse translational motion compensation is performed using 0 SsRA (kr , t) = Ss(kr , t)ei2kr rb0 (t) . (25) 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 6 After the phase correction (25) is applied, a residual phase error caused by the residual spatially invariant translational motion 4π φe (t) = − (r0 (t) − rb0 (t)) (26) λ remains in the (r, t) domain, and the model for the rangealigned signal becomes ssRA (r, t) = Fk−1 {SsRA (kr , t)} r →r Z ∞Z ∞ g(xp , yp )ssp (r, t)dxp dyp . = eiφe (t) −∞ −∞ (27) B. Time window optimization The next part of our ISAR algorithm chooses an optimal slow-time window to be processed in the ISAR image reconstruction. This means locating a slow-time window during which the motion of the object is as smooth as possible. Thus, we need a suitable loss function, whose value quantifies the quality of the result obtained from reconstructing an ISAR image using a certain slow-time window. At this point, there are two difficulties associated with this problem. First, the range alignment procedure compensates for r0 (t) only coarsely, leaving a residual phase error (26) in the range-aligned signal (27). Also, the rotational motion causes spatially variant range cell migration (RCM), which means that scatterers with different initial positions (xp , yp ) each have a unique range history Rp (t) due to the rotational motion. If the range resolution is very high and an equally high crossrange resolution is desired, these effects have to be taken into account when choosing the optimal slow-time window. The basic idea we use is the same as in [44]. We use the contrast of the intensity-normalized ISAR image as the quality measure to determine the optimal length T and position tc for the slow-time window. To obtain the best possible result, the two aforementioned difficulties need to be addressed. This means that the RCM caused by the rotational motion is partially compensated for using keystone formatting [6]. In addition, we use the phase gradient autofocus (PGA) algorithm [20], [21], [48] to remove the spatially invariant phase errors φe in (27). The drawback of introducing these steps into the time window optimization process is that the computational complexity slightly increases. However, by suitably modifying the PGA algorithm, using a sub-optimal interpolation in the keystone formatting, and using a local numerical optimization algorithm to minimize the loss function, reasonable results can be achieved without significantly increasing the computational burden of the original method [44]. In [50], we used similar ideas for determining an optimal slow-time window for the keystone formatting and ISAR image reconstruction. However, the loss function we present here combines the two different parts of the algorithm used in [50]. Essentially, the slow-time window for the keystone formatting and short-time Fourier transform is determined simultaneously in this formulation, reducing the computational complexity. Evaluating the loss function for given values of T and tc proceeds as follows. First, the range-aligned signal ssRA is windowed with a rectangular window of length T centered at tc . Then, the range-Doppler image function t − tc ssRA (r, t) , (28) sS(tc , T ; r, ω) = Ft→ω Π T where ω is the Doppler-variable, is formed. As T and tc are fixed in the following steps, we suppress the dependence of the signal sS (and ss) from them to simplify the notation. Before applying the keystone formatting operation, the residual phase errors φe need to be estimated and removed using an autofocus procedure. Otherwise, the nonlinear part of φe might affect the result in an undesirable way. The standard PGA autofocus algorithm is used here [21], [48], with one simple modification to speed up its convergence. The modification functions to replace the circular shifting operation of PGA by a more efficient procedure. The purpose of the circular shift in the ω-domain is to remove linear phase ramps, which would result in different phase offsets for the derivatives of the phase in each range bin. If the range-Doppler image is severely blurred, this procedure is not effective and multiple iterations of the PGA algorithm are required. This is a problem because multiple iterations increase the computational cost since PGA is used every time the loss function is evaluated. To speed up the convergence, we use an additional procedure proposed originally in [51]. To remove the phase offsets for different range bins, the mean value of the phase of ê in each range bin is calculated as (M −1 ) X h∠ê(r)i = ∠ ê(r, tm ) , (29) m=0 where h·i denotes taking the mean with respect to the slowtime variable and ∠ {·} denotes taking the phase angle of the complex number and ê(r, tm ) = ss(r, tm )ss∗ (r, tm−1 ). (30) In (30), ss stands for the inverse Fourier transform of the range-Doppler image (28), which is windowed symmetrically around the strongest scatterer in each range bin. Subtracting (29) from the phase of (30) produces ê0 (r, tm ) = ê(r, tm )e−ih∠ê(r)i , (31) which removes the effect of different constant phase-offsets for different range bins. After performing the operation in (31), the usual procedure is used to obtain the estimate for the phase error function, namely [21] Z ∞ m X 0 φ̂e (tm ) = ∠ ê (r, tk )dr . (32) k=0 −∞ After obtaining the maximum likelihood estimate (32), the range-compressed signal is phase-corrected as tm − tc ssRA (r, tm )e−iφ̂e (tm ) . ssP GA (tc , T ; r, tm ) = Π T (33) To reduce the computational burden of the optimization procedure, only one iteration of PGA is applied in our algorithm. The purpose of utilizing the procedure above is to make the most of this single iteration. 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 7 After this single iteration of PGA, keystone formatting is performed to produce an intermediate result n ssK (tc , T ; r, τ ) = Fk−1 {Kt→τ {SsP GA (kr , t)}} . r →r (34) n The keystone formatting operation Kt→τ {·} is the change of variables 1/n kc τ , (35) t= kr + kc which in practice is carried out by using a band-limited interpolation technique. The effect of the keystone formatting operation (35) for n = 1 is to remove the linear component of the RCM of every scatterer p simultaneously. This can be deduced by considering the Taylor expansion of the phase of Ssp with respect to t [6]. The ISAR image after PGA and keystone formatting is obtained from sSK (tc , T ; r, ω) = Fτ →ω {ssK (tc , T ; r, τ )}. The intensity-normalized intensity image is defined as ˆ c , T ; r, ω) = R ∞ R ∞ I(tc , T ; r, ω) , I(t I(tc , T ; r0 , ω 0 )dr0 dω 0 −∞ −∞ (36) where I(tc , T ; r, ω) = |sSK (tc , T ; r, ω)|2 . The loss function is chosen so that it quantitatively describes the quality of the ˆ This is a familiar problem in the context of contrast result I. optimization autofocus [14], [22]–[25], [52], and a simple and useful choice is to define the loss function as Z ∞Z ∞ ˆ c , T ; r, ω))drdω, LT W (tc , T ) = ψ(I(t (37) −∞ −∞ where ψ is similar in type to the range alignment problem. The optimal slow-time window is obtained by minimizing the loss function (37). Typically the pulse repetition frequency (PRF) of the radar is high compared to the time scale, in which the motion parameters of the object change significantly. Intuitively, this means that the loss function (37) behaves relatively smoothly, and thus, a computationally efficient local optimization algorithm can be used to find a suitable time window that locally minimizes LT W . The time window optimization process thus produces the optimal window parameters [t?c T ? ]T = arg min LT W (tc , T ). tc ,T The windowed signal is then autofocused using PGA and keystone formatted. After these procedures, the signal is denoted as ssT W . C. Contrast optimization autofocus We proposed using the PGA in the time window optimization due to its simplicity and computational efficiency compared to other possible approaches. However, the contrast optimization autofocus (COA) approach has been demonstrated to produce very good results [14], [22]–[25] and can perform better than PGA or any other autofocus algorithm in certain situations. This superior result is especially true in ISAR imaging, since the object to be imaged typically has at least some strong approximately point-like features, which are emphasized by the loss functions used in the COA. The PGA and COA are in a certain sense complementary. This means that after the optimal time window is selected, autofocused with PGA, and keystone formatted, it can be beneficial to perform COA. We show here that when the residual phase errors after performing PGA are small, COA can be applied in a highly efficient manner. The reason for this is explained in [25]. Stated briefly, the loss function can be regarded as separable when the phase error estimates are relatively small in magnitude. Thus, COA is essentially reduced into a series of one-dimensional optimization problems, which can be solved in parallel. Because the loss function is only approximately separable, typically an iterative procedure is required. In COA, the input signal is phase corrected with an estimate φm of the phase errors to produce an intermediate signal ssC (r, τm ) = ssT W (r, τm )e−iφm , which is used to produce an intensity image I(φ; r, ω) = |sSC (φ; r, ω)|2 , where φ = [φ0 . . . φM −1 ]T and sSC (φ; r, ω) = Fτm →ω {ssC (r, τm )}. When a loss function of the form Z ∞Z ∞ LAF (φ) = ψ(I(φ; r, ω))drdω, (38) −∞ −∞ is used, an expression for the gradient can be obtained from Z ∞Z ∞ ∂ψ(I(φ; r, ω)) ∂I(φ; r, ω) ∂LAF = drdω. (39) ∂φ ∂I ∂φ −∞ −∞ This result is due to Fienup [23], [24], and a straightforward derivation yields Z ∞ ∂LAF =2 Im {ssC (r, τm )M Φ∗ (r, τm )} dr, (40) ∂φm −∞ for the mth component, where ∂ψ(I(φ; r, ω)) −1 sS (r, ω) . Φ(r, τm ) = Fω→τ C m ∂I (41) In Fienup’s original papers [23], [24], a first order numerical optimization algorithm (nonlinear conjugate gradient method) was used to minimize LAF . This kind of approach requires a line search procedure (or alternatively, trust region methods can be used), which increases the computational cost because every time the loss function (38) (or the gradient (40)) is evaluated, a Fourier transform is needed to obtain the rangeDoppler image function sSC . To further increase the computational efficiency of COA, we propose the following method when the residual phase errors are small in magnitude (as they will be after applying PGA). Assuming ψ is twice continuously differentiable, the Hessian matrix can be evaluated and the (m, n) element of it is ∂ 2 LAF = ∂φm ∂φn Z ∞Z ∞ −∞ −∞ ∂ 2 ψ ∂I ∂I ∂ψ ∂ 2 I + ∂I 2 ∂φm ∂φn ∂I ∂φm ∂φn (42) drdω, where a shorthand notation I = I(φ; r, ω) is used. In principle, (42) could be used to solve the autofocus problem with the Newton-Raphson method. However, this is not suitable for two reasons. First, evaluating and inverting the Hessian (42) is numerically a lot more costly than just evaluating the gradient (39). Second, the loss function can be approximated to be a sinusoid in each coordinate direction φm [25], which means 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 8 that far away from the minimum, the Hessian is not positive definite and is close to singular. When the problem is close to being separable (meaning LAF (φ) is approximately a sum of one-dimensional functions), the essential curvature information is in the diagonal elements of the Hessian (42). This means that we can use the one-dimensional Newton-Raphson method (which is also called the secant method) for each phase error component separately. It proceeds iteratively, calculating the phase error estimates and updating them according to k+1 φm = φkm − Gm (φk ) , Hmm (φk ) (43) where the superscripts refer to the number of iteration, Gm = ∂LAF /∂φm and Hmm = ∂ 2 LAF /∂φ2m . The diagonal elements of the Hessian can be obtained by straightforward differentiation, which yields Z ∞ Z ∞ ∂ψ(I) ∂ 2 LAF 2 |ss (r, τ )| = drdω C m ∂φ2m ∂I −∞ −∞ Z ∞ −2 Re {ssC (r, τm )M Φ∗ (r, τm )} dr (44) −∞ Z ∞Z ∞ 2 ∂ ψ(I) +4 · ∂I 2 −∞ −∞ 2 ∗ Im ssC (r, τm )e−iωτm sSC (r, ω) drdω, where the term in last last row can be expanded to avoid unnecessary multiplications in the double integral. In our algorithm, after PGA, a few iterations according to (43) are used to minimize LAF . The signal ssT W (r, τm ) is phase corrected with the obtained estimate as in (33). Denoting φ? = arg minφ LAF (φ), the time windowed, keystone formatted, and autofocused signal is obtained from ? ssAF (r, τm ) = ssT W (r, τm )e−iφm . (45) D. Cross-range scale factor estimation When the ISAR image is reconstructed from the signal ssAF using a Fourier transform with respect to τ , the crossrange dimension of the ISAR image is the angular Doppler frequency variable ω ∈ [−πfP R , πfP R ], where fP R is the sampling frequency in the slow-time direction (which is the PRF of the radar and is assumed to be a constant). To produce an image with a known spatial scale (i.e. the size of the image in spatial units) in both dimensions, a relationship between ω and the cross-range variable y needs to be established. The problem of determining the spatial scale of the ISAR image in the cross-range dimension is referred to as crossrange scaling in the ISAR literature [39]–[43]. Thoroughly understanding the problem requires understanding how the aspect angle θ is related to the cross-range scale of the ISAR image. Re-examining the signal model (9) helps answer this problem. Since the keystone formatting operation is a change of variables (35), it alters the relationship between the slow-time variable and θ. For this reason, we perform crossrange scaling using the range-aligned and autofocused signal which is not keystone-formatted. After the range alignment and autofocus procedures, the spatially invariant phase term is assumed to be compensated for and ideally we have SsAF (kr , t) = Fr→kr {ssAF (r, t)} = G(2kr cos θ(t), 2kr sin θ(t)). (46) According to (46), the autofocused signal SsAF is the twodimensional Fourier transform of g (the ideal ISAR image) evaluated at the points where ky = 2kr sin θ(t) and kx = 2kr cos θ(t) in the two-dimensional spatial frequency domain. Stated another way, in this model, the slow-time variable t actually corresponds to the cross-range spatial frequency variable ky . Thus, the cross-range scale of the ISAR image can be deduced if the sample spacing in the ky variable is known. This follows from applying the Fourier uncertainty relations, which mean that Y ∆ky = 2π, where Y is the cross-range support of the ISAR image and ∆ky is the sample spacing in the cross-range spatial frequency variable. Assuming the signal is relatively narrow-band, we can approximate ky = 2kc sin θ(t). The sample spacing is thus approximately ∆ky = (4π/λ)∆θ(t), where ∆θ(t) is the sample spacing in the aspect angle, which is not necessarily a constant. This means that the cross-range support of the signal is approximately λ . (47) Y = 2∆θ Thus, to determine the cross-range scale of the ISAR image, we need to estimate the aspect angle sample spacing ∆θ. The approach we use to estimate ∆θ is a combination of the rotation correlation [42] and polar mapping [43] methods. They are based on minimizing a certain loss function whose value depends on the estimate for ∆θ. By utilizing two different parts of the signal in the t (ky ) direction, (46) simply means that we are using a rotated version of the Fourier transform G in the image reconstruction. The rotation property of the Fourier transform states that rotating the Fourier transform by an angle θ and inverse transforming back to the spatial domain rotates the original function g by θ. This means that the resulting two images will simply be rotated and scaled versions of each other. If the ISAR image is simply rotated in the spatial domain, the location of the center of the effective rotational motion needs to be known. This leads to the main drawback of the original rotation correlation method [42], which is the additional degree of freedom incurred by this process. However, the location of the center of rotation does not need to be known if the image is rotated in the frequency domain utilizing the rotation and shift properties of the Fourier transform. The sample spacing ∆θ can be estimated by determining the angular shift between two ISAR images that are made from different parts of the signal. The angular shift between the two images is estimated by maximizing the intensity correlation of the two sub-images. Suppose we use two different parts of the signal ssAF (r, t) to reconstruct two distinct ISAR images. The two distinct signals ss1 and ss2 are obtained by applying rectangular windows as t − tcj ssj (r, t) = Π ssAF (r, t), (48) T 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 9 where j = 1, 2, tcj are the time centers of the windows and T is the length of the time window which is obtained by minimizing (37). Next, an estimate for the change of θ between tc1 and tc2 is made. We denote this estimate with θ̂. The second signal ss2 is Fourier transformed with respect to r to produce the signal Ss2 (kr , t) ≈ G2 (kx , ky ). Using (46), we obtain G1 (kx , ky ) = R(θ(tc1 ) − θ(tc2 ))G2 (kx , ky ), (49) where R(θ) denotes the operation which rotates a function by the angle θ. The support of the range spatial frequency variable is kr ∈ [−π/∆r, π/∆r], and the corresponding Cartesian variable is approximated to be kx = 2kr cos θ(t) ≈ 2kr . To rotate the image correctly by utilizing the properties of the Fourier transform, the support of the ky variable has to be known. Using the estimate θ̂, an estimate for the angular spacing is obtained from c = |tc2 − tc1 | θ̂. (50) ∆θ T Consequently, the support of the ky variable is estimated as # " c c 2π ∆θ 2π ∆θ , . (51) ky ∈ − λ λ c is made, it is used in (51). After this, Once the estimate ∆θ the Fourier transform Ss2 = G2 is rotated by the angle θ̂. A highly efficient and accurate way to do this is to decompose the rotation into a series of three shears, which can all be implemented using one-dimensional operations. This idea has been introduced in [53] and elaborated with detail in [54]. The rotated coordinates (kx0 , ky0 ) are obtained using the sequence of transformations 0 ky = kx0 (52) 1 0 ky 1 − tan θ̂2 1 − tan θ̂2 . kx sin θ̂ 1 0 1 0 1 The inverse Fourier transform of the rotated spectrum G2 (kx0 , ky0 ) = R(θ̂)G2 (kx , ky ) produces the rotated and scaled image as n o −1 0 0 g2 (θ̂, x, y) = Fk−1 F G (k , k ) . (53) 2 x y ky →y x →x The loss function to be minimized is the intensity correlation between g2 and g1 , which is defined as Z ∞Z ∞ b x, y)|2 dxdy, (54) LCR (θ̂) = − |g1 (x, y)|2 |g2 (θ, −∞ −∞ n o −1 where g1 (x, y) = Fk−1 F {G (k , k )} , and G1 = 1 x y ky →y x →x ? b Ss1 . Once θ = arg minθb LCR (θ) is obtained, the angular spacing can be deduced using (50) and consequently, the crossrange support is obtained using (47). The above formulation is valid when the sub-images g1 and g2 are made from relatively short slow-time intervals during which the aspect angle θ changes linearly. This method thus only applies if the change in θ is quasi-linear. The advantage of our formulation compared to the polar mapping method [43] is that neither a two-dimensional resampling or a complex iterative procedure is needed. The process that we describe above thus combines the desirable qualities of the rotation correlation [42] and polar mapping [43] methods and overcomes some of their individual shortcomings and limitations. E. Time-frequency imaging approach The final stage in our ISAR imaging algorithm is modifying the ISAR image resulting from the range alignment, time windowing, keystone formatting, and autofocus procedures. This is achieved by replacing the Fourier transform with a high-resolution time-frequency representation. The Cohen’s class is a collection bi-linear time-frequency representations, which are defined as a two-dimensional convolution between a low-pass kernel function and the Wigner-Ville distribution of the signal [55]. An equivalent definition can also be made in terms of the ambiguity function, in which case the twodimensional Fourier transform of the representation belonging to the Cohen’s class is obtained by multiplying the ambiguity function with a suitable kernel function. Several different time-frequency representations have been successfully used in the ISAR image reconstruction [27], [29]– [33], [36], [37], [56]. Our choice is again motivated by simplicity and computational efficiency. Choosing the optimal kernel function for the time-frequency representation is not trivial, and in this case depends on the motion of the noncooperative object. According to our principles, an optimization procedure should be used to determine the parameters of the optimal kernel function, which results in the quantitatively best possible image. The quantitative measure describing the quality of result is again chosen to be the contrast of the intensitynormalized ISAR image. We have previously introduced this methodology in [50]. The above reasoning leads us to use the S-method [57] to produce the final ISAR image. The S-method was first demonstrated in ISAR in [37], where it was shown to surpass the conventional Fourier transform-based image reconstruction by improving blurred and distorted ISAR images due to various target motions. Importantly, the advantages of the Smethod-based approach include e.g. the compensation of highorder phase terms due to both rotational and translational motions and computational efficiency. The simplicity and efficiency choice are based on the fact that, by utilizing the S-method, we can start with the rangeDoppler image sS, which is then used to calculate the Smethod time slice Z ∞ ∗ SM (r, ω) = P (ν)sSAF (r, ω + ν)sSAF (r, ω − ν)dν. −∞ (55) The simplest choice is to use P (ω) = Π (ω/Ω), which means that the kernel of the S-method is described by only one parameter, the length of the integration window in (55). This choice results in Z Ω/2 ∗ SM (Ω; r, ω) = sSAF (r, ω + ν)sSAF (r, ω − ν)dν. −Ω/2 (56) 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 10 To deal with the spatially variant defocusing caused by the rotational motion, we propose using a contrast optimization procedure to determine an optimal range-dependent integration window for the S-method. This means that Ω = Ω(r) is not a constant, but different for each range resolution cell. For this purpose, the loss function to be minimized for range cell r is Z ∞ d (Ω(r); r, ω))dω, LSM (Ω(r); r) = ψ(SM (57) −∞ where d (Ω(r); r, ω) = R ∞ |SM (Ω(r); r, ω)| SM |SM (Ω(r); r, ω 0 )|dω 0 −∞ (58) is the intensity-normalized cross-range profile at range r. The optimal range-dependent integration window Ω(r) is obtained by minimizing (57) for each range cell r. The contrast optimization approach is well-suited for determining the optimal kernel function (in this case Ω) for two reasons. First, if the S-method is able to enhance the resolution of the image in the cross-range direction, then the contrast of the image increases. As the window Ω starts to lengthen, crossterms start to appear between the scatterers, which blur the ISAR image and decrease its contrast. Thus, minimizing LSM will result in a compromise between enhanced resolution and cross-terms. Second, the optimal length for Ω will be relatively short when there are strong scattering centers nearby. Since ω is in reality a sampled variable, a very small amount of computation is thus needed for minimizing LSM . We note that the slow-time window optimization procedure presented in Section III-B essentially performs a part of determining the kernel function of the time-frequency representation. When using the S-method, only a single time sample (ISAR image frame) of the time-frequency representation can be calculated very efficiently using (56). A further procedure which can be used to enhance the contrast of the ISAR image is the time-frequency reassignment method [58]. We have used this procedure in a previous version of our algorithm [50], and its use in ISAR has been demonstrated with experimental data in [59], [60]. However, the procedure we use in this paper automatically locates a slow-time interval during which the motion of the object is as smooth as possible. So, while the reassignment method possibly increases the contrast slightly, the increase in image quality does not warrant the extra computation required to calculate the reassigned image function. Furthermore, if the reassignment operation is performed, it can be done only in the frequency direction ω, since the full time-frequency representation is not calculated in our ISAR algorithm. Fig. 2 presents a summary of the ISAR processing. It also briefly describes the proposed improvements, which are decomposed into several successive steps for clarity. IV. N UMERICAL RESULTS A. Experimental setup We demonstrate the ISAR algorithm presented in Fig. 2 with a numerical example. The data we use were obtained using a turntable and a radar system with a stepped-frequency waveform with horizontal polarization. For benchmarking our algorithm, we produced a noncooperative scenario with objects having hard and unknown motion states. We induced artificial translational motions in the turntable data, after which we resampled the data in the slow-time direction to produce a nonlinear change in the aspect angle θ. The purpose of this experiment is to test challenging motion compensation scenarios and demonstrate the performance of our ISAR algorithm using real data while knowing the ground truth of the motions. Then we are able to compare the obtained results with the ground truth. The measurements were conducted at Ylöjärvi, Finland, at an open space measurement range, approximately 240 meters away from the target and at an elevation angle of 0 degrees. The hardware used in the measurement campaign included a radar manufactured by System Planning Corporation, USA, and a turntable manufactured by Dynaset Oy, Finland. We used five measurements of different objects. The measured vehicles can be seen in Fig. 3. The length of the slowtime window for the range-compressed signal prior to ISAR processing was chosen to be 1000 samples corresponding to an integration time of approximately 30 seconds. The burst repetition frequency of the radar was approximately 35 Hz and the objects rotated approximately 60 degrees during the chosen slow-time interval. The frequency bandwidth of the waveform was B = 1 GHz producing a range resolution of δr = 0.15 m, while the carrier frequency was set to fc = 8.5 GHz. However, for our noncooperative scenario, this data was resampled so that the data follows the artificial motions with an adequate pulse repetition frequency avoiding cross-range ambiguities. For each vehicle, four different trajectories were emulated. The range trajectories were polynomials of randomly chosen order (between second and fifth) with random uniformly distributed coefficients. The range trajectories were normalized to make the difference between the maximum and minimum values of r0 slightly smaller than the unambiguous range window of 70 meters. The angular trajectories were emulated similarly, the only difference being that the polynomials were chosen to be monotonically increasing. As an example, both the translational and rotational motions emulated for the fifth vehicle are shown in Fig. 4. Altogether we have 20 different target trajectories for the subsequently considered tests. B. Results Referring back to Fig. 2, our ISAR algorithm starts with the range alignment procedure. We used the steepest descent method with a fifth order Legendre polynomial parametrization and a line search procedure to solve it. The progress of the numerical optimization algorithm is demonstrated for the mean squared envelope difference loss function in Fig. 5, which shows the loss values and gradient norms after each iteration. The quality of the estimation result can be analyzed by calculating the standard deviations of the residual error re (t) = r0 (t) − rb0 (t) as q 2 σe = h(re (t) − hre (t)i) i. 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 11 TABLE I: Mean residual errors for the different initial guesses and loss functions. The loss functions L1 and L2 are the benchmarks, while L1 + L2 and L1 L2 are the proposed new loss functions. Loss function / Initial guess (14) (15) L1 L2 L1 + L2 L1 L2 σe 0.5313 0.3044 0.1674 0.1512 0.1076 0.0979 Ideally, this value should be smaller than half the range resolution δr. We compared the performance of two different initial guesses and four different loss functions in terms of the residual errors. The initial guesses we compared were the center of mass (14) and the maximum correlation (15), and the loss functions we considered were the entropy of the sum envelope (L1 ), the mean squared envelope difference (L2 ), and the two combinations which are the sum L1 + L2 and the product L1 L2 of these two. The results are presented in Table I. The maximum correlation method was more accurate as the initial guess and the product LRA = L1 L2 was slightly more accurate than the sum, although the difference between them was only one centimeter. As seen from the results in Table I, the proposed loss functions surpassed the benchmarks by 35 percent in terms of the residual estimation error. This validates the partially heuristic justification made in Section III-A, where we introduced these new loss functions. The intensity of the range-aligned signal for one trajectory realization after using the obtained estimate to align the range profiles is shown in Fig. 6 (b). After the range alignment, the optimal slow-time window was located by minimizing LT W . In this part of our ISAR algorithm, it is of interest to compare the result with the maximum contrast time window optimization procedure in [44]. In [44], an image contrast-based autofocusing—essentially the same algorithm as COA—is performed prior to evaluating the loss function of negative image contrast. The most important factor in this comparison is the motion compensation included in the loss evaluation. To ensure that the results of different motion compensation strategies were comparable, we used the same optimization algorithm for all of them. The optimization algorithm we used was a simple coordinate descent search, where one-dimensional line searches are performed successively in different coordinate directions. This optimization algorithm is slightly more computationally intensive than the one used in [44] but it also avoids the need for choosing the threshold value included in [44], to which the algorithm is sensitive and which can significantly affect the length of the time-window. The results for four different motion compensation strategies in the time-window optimization are shown in Table II. As seen from these results, the strategy of including the keystone formatting is computationally most efficient and produces the best image contrast value with the PGA. However, the inclusion of the keystone formatting increased the contrast by 6–10 percent compared to both benchmarks, COA and PGA. Additionally, the proposed strategy of using TABLE II: Mean values for the optimal window lengths, loss values, and relative computational speeds for different motion compensation strategies in the time-window optimization. The rows COA and PGA are the benchmarks. Motion compensation COA COA + Keystone formatting PGA PGA + Keystone formatting T? 336 352 320 363 Image contrast 77.57 82.55 90.30 99.89 Relative speed 1 0.79 7.5 4.8 TABLE III: The image contrasts for three different cases: the range-Doppler image without the S-method modification (L = 0), the S-method image with a fixed window length L (benchmark), and the S-method image with a range dependent window length L(r). S-method window L=0 Fixed L Range-dependent L = L(r) Image contrast 99.89 121.34 128.73 the PGA in conjunction with the keystone formatting was almost five times faster than the benchmark based on COA alone. After the optimal slow-time window was located, the signal was autofocused using PGA, keystone formatted, and autofocused using COA. In our example, the COA consisted of a single iteration of (43). A single iteration sufficed because the residual phase errors after performing PGA were very small in magnitude. After these procedures, the S-method kernel optimization and cross-range scaling procedures followed. The optimal integration window for the S-method was obtained by minimizing the entropy of the S-method cross-range profiles. The entropy of the cross-range profile containing the highest total energy for the first vehicle and trajectory realization is shown in Fig. 7 (a). In Table III, the resulting image contrasts for a fixed integration window L and a range-dependent L are presented alongside with that of the range-Doppler (Fourier-transform-based) image (L = 0). As seen in Table III, our modification increased the image contrast by six percent compared to a fixed L. The cross-range scaling problem was solved by dividing the optimal slow-time window into two non-overlapping parts. Then the second part was rotated in the spatial frequency domain, and the rotational angle which minimizes the intensity correlation loss function (54) (shown in Fig. 7 (b)) was used to evaluate the cross-range scale of the ISAR image. This concluded our ISAR algorithm, and the obtained result for each vehicle and one of the trajectories are shown in Fig. 3 on homogeneous two-dimensional Cartesian coordinate grids in the spatial domain. For reference, we implemented an ISAR algorithm using the same well-established optimization steps that we use but without the modifications proposed in this paper. The result for the first vehicle and trajectory realization without and with our modifications are shown in Figs. 8 (a) and (b), respectively. For both images, we calculated a widely used quality measure used to assess image contrast, the ratio between the standard deviation and the mean of the image intensities. As can 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 12 be seen, the use of the proposed improvements produced an ISAR image with a notable 50 percent higher contrast value than that in the reference for this trajectory realization. Taking into account all the 20 different use-cases, the image contrast increased by 28 percent on average. Less spatially variant blurring also occurs. It should be noted that the result in Fig. 8 (b) represents one of the best outcomes of the performed experiments. When the motion of the object was significantly simpler due to the randomly emulated trajectory, the proposed optimization steps were naturally not able to increase the image contrast as much as above. Thus, the advantages of the proposed improvements are the greatest when the noncooperative object exhibits changing translational and rotational motions during the ISAR data acquisition. The results of well-focused ISAR imaging in Fig. 3 contain plenty of useful information for noncooperative target recognition (NCTR) or automatic target recognition (ATR) purposes. Importantly, the size, the shape, and the dominant scatterers of the objects in Figs 3 (f), (g), (i), and (j) are clearly distinguishable because of the high quality of the motion compensation and cross-range scaling. A more difficult case is presented in Fig. 3 (h), where the aspect from the front of the car makes it more challenging to determine the shape or the size of the object. Notice that the color scalings in Figs. 3 (f)–(j) are based on the maximum image intensity. This makes some of their features harder to interpret visually. In many current NCTR and ATR approaches, major challenges are related to image formation due to clutter and the unsolvable motions of the imaged object. Another challenge is the problem of limited training data concerning all the necessary object classes, radar parameters, and aspect angles. Overcoming these challenges is the subject of current NCTR/ATR research but is out of the scope of this paper. C. Computational load analysis To consider the computational load of our ISAR algorithm, we evaluated the computational cost of each stage of the algorithm using the analysis and expressions we derived for the different loss functions and their derivatives. The order of magnitude and relative costs between the different parts of the algorithm are of interest rather than the exact number of arithmetic operations. Table IV shows approximations (which are used to evaluate the correct order of magnitude) for the total number of arithmetic operations required for evaluating each loss function or its derivative. In Table IV, M is the number of samples in the slow-time direction, and N is the number of samples in the range direction. In this analysis, we have not taken into account the fact that the number of samples M in the slow-time direction changes in different parts of the algorithm. However, we assume that the order of magnitude of the number of samples does not change after the time window optimization procedure, in which case the values in Table IV provide a useful comparison. The computational costs are similar for all loss functions except for the derivatives of LAF , which are included in Gm /Hmm . However, to calculate the total cost we need to take into account how many times each of the expressions TABLE IV: The approximate number of arithmetic operations required for evaluating each expression. Expression LRA ∂LRA /∂∆r Operation count M N log N M N log N LT W LAF Gm /Hmm M N log M 2 N M N log M M N (log M + M ) LCR M N log N 2 M 3 3 Relative cost (M = N ) 0.03 0.03 0.08 0.03 1 0.08 in Table IV is evaluated in our ISAR algorithm. Since COA is performed after PGA to remove the residual phase errors, Gm /Hmm is evaluated only once in our ISAR algorithm; every other loss function and derivative in Table IV is used in a local optimization algorithm, which includes a line search procedure. Considering this and the numerical results of this Section, we see that these loss functions need to be evaluated far more often. Typically the algorithms are set to run a maximum of ten iterations, and each iteration includes a line search procedure with a maximum of ten evaluations of the loss or gradient function. In conclusion, the computational costs of the individual parts of the ISAR algorithm are actually nearly identical. To illustrate the computational load of our ISAR algorithm, the algorithms were implemented using MATLAB and tested using a laptop with a 2.60 GHz dual-core processor and 8 GB random access memory. In the experiments performed in this Section, our algorithm took about 25 seconds to complete on the average. The most time-consuming part was the timewindow optimization procedure, which took approximately half of the total computation time. The ISAR algorithm using the same well-established optimization steps without our modifications averaged at 45 seconds. The most significant speedup was achieved in the range alignment phase of the ISAR algorithm, where our modification increased the computational speed by an order of magnitude. This can be attributed to the fact that we use a carefully selected initial guess and first order optimization, whereas in the original method [15] black box numerical optimization was used. Regarding the time-window optimization procedure, the computational speeds of different motion compensation strategies were listed in Table II. These results show that using PGA and keystone formatting rather than COA was five times faster. The comparison between the computation times should be considered only as indicative, since all the optimization steps in Fig. 2 have not been considered together as a wellestablished approach before. Moreover, the exact implementations of the optimization algorithms are not fully described in all the references, which presents a challenge for a meaningful comparison of the overall ISAR algorithm. Also, as seen from Table II, the motion compensation strategy used in the time window optimization significantly affects the computation time (PGA was used in these experiments). However, considering the results of the performed tests, the proposed improvements resulted in increased computational efficiency in the ISAR processing overall. 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 13 D. Summary of the results The key results of this paper can be summarized as follows: 1) The computational cost of the global range alignment method was reduced by an order of magnitude, and the estimation performance was enhanced by 35 percent (see Table I). These results were achieved by using an initial guess for the range shifts (14), deriving an expression for the gradient of the loss function (19) and using combinations of the loss functions as in (22). 2) Including keystone formatting in the time-window optimization procedure increased the image contrast by 6–10 percent compared to the benchmarks. In addition, the proposed improvement performed the motion compensation and loss evaluation five times faster than COA. 3) A new expression for the second order partial derivatives of the loss function used in COA was derived in (38)– (44). This expression provides a practical solution to the long-standing contrast maximization problem typically involving heuristic numerical optimization. In particular, the solution not only reduces the computational complexity but also leads to simple algorithmic implementation, as discussed in Section III-C. 4) The rotation correlation method, which fundamentally assumes a known center of target rotation, and the polar mapping method, which relies on a two-dimensional resampling operation, are typically used separately to solve the cross-range scaling problem. However, since the two methods rely on a cost function of equivalent nature, see (54), combining them was empirically validated to yield a low discrepancy of 4–8 percent in terms of the true width of the imaged objects. It is noteworthy that no knowledge of the target center was required, and only computationally efficient one-dimensional interpolations were used. 5) The proposed contrast optimization procedure, which determines the optimal kernel for the time-frequency representation per range bin (55)–(58), enhanced the contrast of the ISAR image reconstruction by six percent compared to the traditional S-method approach with with a fixed length kernel. Evidently, the spatially variant blurring caused by the nonuniform rotational motion of the car was significantly reduced, as shown in Fig. 8. In this way, our modifications to the optimization-based data-driven motion compensation techniques were shown both to increase the computational efficiency and image quality. V. C ONCLUSION We have described in detail how optimization can be used in every part of data-driven motion compensation and ISAR image reconstruction. Firstly, the performance of the global range alignment method was enhanced by combining previously suggested loss functions and utilizing first order numerical optimization. After that, the time window optimization process was enhanced by performing autofocus and keystone formatting before evaluating the contrast of the ISAR image. Then, the contrast optimization autofocus approach was utilized by deriving an expression for the second order partial derivatives of the loss function. The time-frequency based imaging approach was also enhanced by choosing the optimal kernel for the time-frequency representation based on the image contrast. Finally, the cross-range scaling problem was solved by combining the rotation correlation and polar mapping methods. By combining all the proposed findings, a computationally efficient ISAR algorithm was demonstrated to improve the imaging performance under complicated target motion dynamics. 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Venkataramaniah, “ISAR imaging of moving targets based on reassigned smoothed pseudo wigner-ville distribution,” in Proceedings of International Radar Symposium India, IRSI-2011, 2011. 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 15 Risto Vehmas (S’17) is a Ph.D. student at the Tampere University of Technology (TUT) in Tampere, Finland. He received his M.Sc. degree in theoretical physics from the University of Oulu, Oulu, Finland, in 2014. He has been working at the Department of Signal Processing at TUT since 2015 on topics related to synthetic aperture radar signal processing. Juha Jylhä is a researcher and project manager in industrial projects at Tampere University of Technology. He received his M.Sc. degree in electrical engineering from Tampere University of Technology, Tampere, Finland, in 2005. His current research interests include system modeling, signal processing, and artificial intelligence in remote sensing and military applications. Jylhä is a candidate for a Ph.D. in computing and electrical engineering at Tampere University of Technology. Ari Visa (M’79, SM’96) received the M.S. degree in computer science and technology from the Linköping University of Technology, Linköping, Sweden, in 1981, and the Doctor of Technology degree in information science from Helsinki University of Technology (Aalto), Espoo, Finland, in 1990. He is also a Docent in image analysis at Aalto University. Dr. Visa is currently a Professor in Digital Signal Processing and the Vice Dean of the Faculty of Computing and Electrical Engineering at Tampere University of Technology. From 1981 to 1983, he was with the Linköping University of Technology as a Research and Teaching Assistant in the Computer Vision Laboratory. He was involved in several image processing projects. From 1984 to 1985, he was employed by the Finnish Research Centre, as a researcher in the field of image analysis in paper quality control. From 1985 to 1988, he was a senior member in technical staff in industrial applications of image analysis at Jaakko Pöyry Corporation. From 1988 to 1993, he was a senior researcher and a project manager in the field of image analysis with neural networks at the Finnish Research Centre. In 1993, he joined the laboratory of Computer and Information Science. From 1996 to 2000 he was professor in Computer Science at Lappeenranta University of Technology. Currently, Dr. Visa is a professor in Signal Processing at Tampere University of Technology. He is co-author to more than 200 refereed international papers. He holds several patents. He is active in many conference program committees and evaluates research papers and proposals. Dr. Visas current research interests are in multimedia, adaptive systems, wireless communications, distributed computing, soft computing, computer vision, knowledge mining, and knowledge retrieval. Dr. Visa is the former President and Vice President of the Pattern Recognition Society of Finland and the former chairman of IAPR Workgroup TC3 Machine Learning. Minna Väilä is a researcher at the Tampere University of Technology in Finland. She received her M.Sc. degree in signal processing from the Tampere University of Technology, Tampere, Finland, in 2010. She is currently a doctoral student in computing and electrical engineering at Tampere University of Technology. Her research interests are in topics relating to radar, including target recognition, radar response simulation, and performance modeling. Juho Vihonen (M’15) received his M.Sc. in automation engineering and D.Sc. (Tech.) in signal processing from Tampere University of Technology (TUT), Finland, in 2003 and 2009, respectively. From 2001 to 2017, he was with the Department of Signal Processing (DSP), TUT, where he held the position of a Senior Research Fellow. Currently, he is with Cargotec, where his responsibilities have focused on AI capabilities and analytics architectures for digital services. Cargotec is a leading provider of cargo and load handling solutions. His research interests lie in the fields of machine learning and real-time processing of multi-sensor signals for control of heavy-duty robotic manipulators. 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 16 Fig. 1: Illustration of a two-dimensional imaging geometry for ISAR for a fixed instant of slow-time. The primed coordinate system is rigidly attached to the object which moves in the unprimed system. The coordinate systems are related by a shift and a rotation about the axis perpendicular to the (x, y)-plane. 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 17 Fig. 2: State-of-the-art ISAR processing decomposed into algorithmic steps. The novel contributions proposed are indicated below the steps. In a typical use-case, for example, the time window is kept fixed without optimization. 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 18 Fig. 3: The noncooperative objects used in the ISAR experiments (a)–(e) and the ISAR images of them resulting from our ISAR algorithm (f)–(j). (a) The emulated translational motions for the fifth vehicle. (b) The emulated rotational motions for the fifth vehicle. Fig. 4: The artificially induced translational (a) and rotational (b) motions of the car in the noncooperative scenario presented as functions of slow-time t. Note the introduced accelerations and decelerations, which present a challenge for any data-driven motion compensation. Fig. 5: The loss values (left) and gradient norms (right) after each iteration in the range alignment problem. Note the steep reduction of the errors just after a few iterations, attributing to computational efficiency, as discussed. 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAES.2017.2756518, IEEE Transactions on Aerospace and Electronic Systems 19 (a) Before range alignment. (b) After range alignment. Fig. 6: The intensity of the range-compressed signal (a) before and (b) after range alignment. Without the range alignment, the translational motion in (a) would impair the image reconstruction. By compensating for the translational motion, the best result in Table I produces a smooth, accurate outcome, as shown in (b). (a) The entropy of the S-method cross-range profile. (b) The graph of the intensity correlation loss function (54). Fig. 7: As can be seen in (a), the optimal length of the S-method window is four samples. Thereby, to compensate for the blurring caused by the rotational motion, only a small amount of computation is required. Recall also that the cross-range scale of the ISAR image is determined by the minimum of LCR in (b), which is easy to evaluate because of the smoothness of the graph. (a) ISAR image obtained without our algorithmic improvements. (b) ISAR image obtained with our algorithmic improvements. Fig. 8: The result of an ISAR algorithm without our improvements for the noncooperative object Ford Focus depicted in Fig. 3 (a) is shown in (a), while the result of our ISAR algorithm is shown in (b). Note the improved contrast value in (b). 0018-9251 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.