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Holism-Based Features for Target Classification in Focused and Complex-Valued Synthetic Aperture Radar Imagery

often-ignored phase chip. The second framework uses the complex-valued 2-D synthetic aperture radar chip after it is transformed into a 1-D vector. Representative features are introduced under each framework. Further, for comparison purposes, baseline features from the power-detected chip are also considered. Three feature sets are extracted from the real-world MSTAR data set and used separately and combinatorially to design multiple instances of an eight-class support vector machine classifier. A classification accuracy of 93.42% is achieved for the holism-based features. This is in comparison to 73.63% for the baseline features. Using Fisher scoring to measure the information contained in each feature, top-ranked features from the first and second holism-based frameworks, respectively, are found to be 7 and 160 times those of the baseline features. Because the nonlinear phenomenon is resolution dependent, our proposed approach is expected to achieve even greater accuracy for synthetic aperture radar sensors with higher resolution.

KHALID EL-DARYMLI, Member, IEEE Northern Radar Inc. St. John’s, Canada PETER MCGUIRE, Member, IEEE C-CORE St. John’s, Canada

I. INTRODUCTION

ERIC W. GILL, Senior Member, IEEE Memorial University of Newfoundland St. John’s, Canada DESMOND POWER, Member, IEEE C-CORE St. John’s, Canada CECILIA MOLONEY, Member, IEEE Memorial University of Newfoundland St. John’s, Canada

Reductionism and holism are two worldviews underlying the fields of linear and nonlinear signal processing, respectively. Conventional radar resolution theory is motivated by the former view, and it is violated by nonlinear phase modulation induced by dispersive scattering typically associated with extended targets. Motivated by the latter view, this paper offers a new insight into the process of feature extraction for target-recognition applications in single-channel imagery output from synthetic aperture radar processors. Two novel frameworks for holism-based feature extraction are presented. The first framework is based solely on the Manuscript received October 10, 2014; revised April 1, 2015; released for publication October 19, 2015. DOI. No. 10.1109/TAES.2015.140757. Refereeing of this contribution was handled by P. Lombardo. This work is supported in part by the Research and Development Corporation of Newfoundland and Labrador, the Atlantic Innovation Fund, and the Natural Sciences and Engineering Research Council of Canada. Authors’ addresses: K. El-Darymli, Northern Radar Inc., 25 Anderson Ave., St. John’s, Newfoundland, Canada A1B 3E4; P. McGuire, D. Power, C-CORE, Bartlett Building, Morrissey Rd., St. John’s, Newfoundland, Canada A1C 3X5; E. W. Gill, C. Moloney, Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada A1B 3X5. Corresponding author is K. El-Darymli, E-mail: ([email protected]). C 2016 IEEE 0018-9251/16/$26.00

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The underpinning philosophy for science in general, and the field of signal processing in particular, is based on either one of two interdisciplinary worldviews: reductionism (also known as Newtonianism) and holism [1–6]. In the reductionist worldview, a complex system is assumed to be simply the superposition of its parts, and its analysis is reduced to the analysis of its individual components. Although this view may not seem to explicitly dismiss the existence of the so-called emergence phenomenon (i.e., multiplicity due to interactions between the individual components), it is implicit that the emergence phenomenon can be captured by the constituent processes. On the contrary, in the holistic worldview the system is viewed as a whole that cannot be fully understood solely in terms of its constituent parts. This principle was succinctly summarized 24 centuries ago by Aristotle in Metaphysics: “The whole is more (or other) than the sum of its parts” [1, 3, 6]. Reductionism and holism set the philosophical foundations of linear and adaptive/complex-valued/ nonlinear/nonstationary signal processing, respectively [1, 2, 4, 6]. In linear system theory [7], the reductionist view is applied, meaning that the signal is decomposed into fragments that are analyzed individually. The analysis result for the whole signal is obtained from proper scaling (i.e., the homogeneity property) and addition of the fragments (i.e., the superposition principle). For this process to be valid, the central limit theorem is invoked; hence, it is implicitly assumed that the signal samples are drawn from a distribution possessing a finite variance [8]. Accordingly, linear system theory treats deviation from linearity as noise that warrants removal. For example, the Fourier view, the heart of linear system theory, assumes a first-order fundamental oscillation and bounding higher order harmonics. Despite its mathematical soundness, this view does not correspond strictly to physical reality [9]. When the underlying random processes are nonlinear, advantages of the holism-based approach become apparent. Statistically, nonlinear signal processing is

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 52, NO. 2 APRIL 2016

motivated by the generalized central limit theorem, which holds that the variance of the underlying random variables is infinite [10]. The Poincaré view [9, 11] is one such important view for nonlinear signal processing, which sets the foundations for chaos theory. The Hilbert view [9] is another important view, which was popularized after the advent of the Hilbert–Huang transform [12], an important advancement in adaptive, nonlinear, and nonstationary signal processing. Conventional radar-resolution theory, which is a resolution theory of point targets [13], represents a direct application of the reductionist worldview. Thus, analysis of the single-channel synthetic aperture radar (SAR) imagery output from SAR processors, for target-recognition applications, has traditionally been based on techniques associated with the image intensity, ignoring the phase content [14, 15]. The insufficiency of linear resolution theory to extended targets, based on the empirical observation that nonnatural targets produce dispersive scattering, has been reported in the literature [16–18]. In effect, this induces a nonlinear phase modulation in the radar return signal which causes a mismatch in the correlator’s output. This phenomenon is preserved in the complex-valued image rather than the detected one. In [16–18], an algorithm is proposed to recognize nonlinear scattering in complex-valued imagery output from SAR processors. The approach followed aims at tracking phase-center shifts due to the nonlinear phase modulation induced by dispersive scattering, and it employs the assumption that the nonlinear response in the complex-valued SAR image is due to a composite interaction between two or more point scatterers in the extended target. However, there are two possible drawbacks associated with this approach. First, it is nonadaptive due to its contingency on certain a priori assumptions, as mentioned earlier. Second, while it identifies nonlinear scattering in general terms, it neither classifies it nor estimates its order. The holism-based feature-extraction approach presented in this paper differs significantly in that it is entirely data driven without any a priori assumptions. Further, our approach is advantageous in that it allows for classifying the dispersive scatterers as well as estimating their nonlinear order. To our knowledge, this capability has not been previously demonstrated in the SAR literature. This provides for significant use of nonlinear phenomena in target-recognition applications. The study in this paper builds on six previous investigations [19–24]. In [19, 20], an in-depth analysis for nonlinearity in single-channel SAR imagery was conducted. The analysis demonstrated the statistical significance of the nonlinear phenomenon in high-resolution complex-valued SAR imagery. It was also shown that the nonlinear effect is obliterated or diminished, respectively, for magnitude and power detections. In [21–23], a method for characterization and statistical modeling of the phase in single-channel SAR

imagery was proposed. Also, the circularity (also known as propriety) in complex-valued SAR imagery was investigated. It was demonstrated that, for the case of an extended target, the complex-valued SAR chip is noncircular. In [24], a method for estimating the order of nonlinear dispersive scattering in complex-valued SAR imagery was provided. The main contributions of this study are as follows: • Development of a new feature set based solely on the phase in single-channel SAR imagery. • Development of three complementary 1-D representations for 2-D real and imaginary parts and bivariate and complex-valued SAR chips. This allows for application of various holism-based methods for feature extraction. • Development of a new set of features based on the Hilbert–Huang transform as well as methods motivated by chaos theory, including permutation entropy and detrended fluctuation analysis. • Development of a simple method for feature standardization based on the median and the interquartile range.

The topic addressed in this paper is applicable to various kinds of stationary and moving targets, including vehicles, ships, airplanes, icebergs, oil slicks, and more. Furthermore, the application of the methods proposed may be extended beyond SAR to include radar in general, sonar, synthetic aperture sonar (SAS), ultrasound, synthetic aperture ultrasound (SAU), and so on. The remainder of this paper is organized as follows. First, the origin of the nonlinear phenomenon in single-channel SAR imagery is discussed in Section II. Then the proposed framework for feature extraction based solely on the phase chip is described in Section III. Third, the three 1-D representations for the 2-D SAR chip are introduced in Section IV. Subsequently, methods used for feature extraction based solely on the 1-D representations are proposed in Section V, and this is followed in Section VI by a discussion of the SAR data set utilized in this study. The overall features used in the study, including the baseline and holism-based (i.e., phase-based and 1-D-based) features, as well as the proposed procedure for feature normalization, are defined in Section VII. The processes of classifier design and feature selection are outlined in Section VIII, and the overall results are elaborated upon in Section IX. Finally, conclusions are offered in Section X. II. ORIGIN OF THE NONLINEAR PHENOMENON IN SAR IMAGERY

The baseband backscatter xBB from a single point target, outputted from the quadrature demodulator and downlinked to the SAR processor, is known as the phase history or the raw data and is given by [13]

EL-DARYMLI ET AL.: HOLISM FOR TARGET CLASSIFICATION IN SYNTHETIC APERTURE RADAR IMAGERY

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xBB (τ, η) = A exp (j ψ) wr

R (η) R (η) 2 R (η) , (1) τ −2 wa (η − ηc ) exp −j 4πfo exp j πKr τ − 2 c c c

where A is the backscatter coefficient (i.e., σ o ), ψ is a phase change in the received√ pulse due to the scattering process from a surface, j = −1, τ is the fast time, η is the slow time, wr (τ ) = rect(τ /Tr ) is the transmitted pulse envelope, Tr is the pulse duration, R(η) is the distance between the radar and the point target, c is the speed of light in a vacuum, wa (η) is the two-way azimuth beam pattern, ηc is the beam center in the azimuth direction, fo is the center frequency, and Kr is the frequency-modulation rate of the range pulse. The SAR raw signal xBB (τ, η) is conventionally modeled as xBB (τ, η) = g (τ, η) ⊗ h (τ, η) + n (τ, η) ,

(2)

where ⊗ denotes convolution, g(τ , η) is the ground reflectivity, h(τ , η) is the impulse response of the SAR, and n(τ , η) is a noise component mainly due to the front-end receiver. The SAR processor solves for g(τ , η). Following conventional radar-resolution theory, h(τ , η)—bounded by the curly brackets in (1)—is an impulse response of a point target. For a given reflector within the radar illumination time, ψ is assumed to be constant [13]. For the case of an extended target, this assumption is adopted verbatim. Such a target is thus modeled as the linear combination of its point reflectors. However, the assumption of constant ψ is violated in the presence of dispersive scattering from cavity-like reflectors, typical in stationary and moving nonnatural (extended) targets such as vehicles and airplanes. These reflectors trap the incident wave before it is backscattered, thus inducing a phase modulation (PM). The problem arises when the PM is nonlinear. Besides the PM, this phenomenon also introduces amplitude modulation [16–18]. Therefore, the backscatter term in (1) is rewritten as s (τ (fτ ) , η) = A (τ (fτ ) , η) exp (j ψ (τ (fτ ) , η)) ,

where O is the order of nonlinearity induced by the dispersive scatterer. For O ∈ {0, 1, 2}, the PM is linear and its effect is either translation or smearing of the response in the correlation filter. Another reason for the smearing of the response is the variable Doppler processing used for motion compensation. However, for O ∈ / {0, 1, 2}, the phase center possesses a nonlinear delay which introduces spurious effects in the correlator’s output. This phenomenon is referred to as sideband responses, and the information about it is preserved in the complex-valued image rather than the detected one. Further, in the presence of an extended target, it is empirically observed that this effect dominates the focused SAR imagery [16–18]. The sideband responses are radically different from the range and Doppler side lobes. One of the reasons for this is that they are among the strongest responses. Second, unlike the range and Doppler side lobes, the sideband responses are not restricted to the range and cross-range gates. Third, they are distributed over an area far larger than that occupied by the target. As stressed in [16–18], these sideband responses cannot be suppressed by the weighting methods because they are target generated. Obviously, the nonlinear PM violates the resolution theory of point targets, which advocates discarding the phase content in the complex-valued image output from SAR processors. III. ADAPTIVE STATISTICAL MEASURES BASED SOLELY ON THE PHASE CHIP

In previous work [21, 22], it has been demonstrated that the phase in single-channel complex-valued SAR imagery, particularly in the presence of extended targets, can indeed be statistically well modeled using the wrapped complex Gaussian scale mixture (WCGSM). A brief overview of our proposed algorithm for phase characterization is presented in Fig. 1. The

(3)

where τ (fτ ) is the time delay due to the PM and fτ varies over the spectral width B of the chirp. It should be emphasized that the magnitude and phase of the backscatter in (3) are frequency dependent. While the amplitude modulation is a linear process, this is often not the case for the PM. Indeed, based on the principle of stationary phase, the time delay induced by a dispersive scatterer is τ (fτ ) ∝ 788

d (fτ )O , df

(4) Fig. 1. Proposed algorithm for phase characterization in SAR imagery.


first converted to the polar form1 as follows: CVF (u, v) = exp (j (x, y)) = cos { (x, y)} + j sin { (x, y)} .

(7)

CVF(u, v) has a size of Mc × Nc . The convolution kernel fk has a size of Mk × Nk . The values of Mk and Nk are typically chosen to be odd, to avoid ambiguity in defining the center pixel. Convolving CVF(u, v) with the kernel fk at a particular pixel location (mc , nc ) in the chip yields the convolution image as CI (mc , nc ) = fk ⊗ CVF (mc , nc ) =

Nk Mk

fk (mk , nk ) CVF (mc − mk , nc − nk ),

mk =1 nk =1

(8) Fig. 2. Three convolution kernels (fk ) considered in this study.

complex-valued SAR chip is available in the form c (u, v) = i (x, y) + j q (x, y) ,

(5)

where i(x, y) and q(x, y) are the real and imaginary parts, respectively. Note that (x, y) represents the 2-D Cartesian coordinates in the real-valued plane, while (u, v) represents the 2-D Cartesian coordinates in the complex-valued plane. Hence, the phase chip is given by (x, y) = arg {c (u, v)} .

(6)

The phase chip (x, y) is processed in order to make sense of the information content it carries. This is because, by definition, the phase is always relative, and it often appears meaningless if it is not appropriately characterized. Accordingly, the proposed algorithm simply produces the so-called backscatter relative phase image (BRPI). The main idea is that each pixel in the phase chip is characterized in relation to its neighbors. Then a histogram can be computed based on the BRPI. The BRPI is computed from the difference between the phase chip and the neighborhood-processed phase chip. The latter is obtained through convolving the phase chip with a kernel fk . The kernel should have a value of 0 in the center and 1 where desired. The kernel convolution operation produces an average phase chip for the neighborhood of each pixel in the phase chip. In this study, three kernels are considered (Fig. 2). As pointed out in [21], these kernels were chosen because they are found to produce histograms consistent with typical statistical models. Phase values are in the range (−π, π]—that is, the phase values are on a circle where the angles π and −π meet at the same point. In order to account for the circularity of the phase, the phase chip (x, y) should be

where ⊗ denotes convolution, mc ∈ {1, 2, . . . , Mc }, and nc ∈ {1, 2, . . . , Nc }. The BRPI is defined as the difference between the phase angles pertaining to the original phase chip and the convolution chip, and may be expressed as

CVF (u, v) BRPI (x, y) = arg . (9) CI (u, v) The resultant BRPI defines the characteristic phase of each pixel in the input phase chip (x, y) relative to its neighborhood as defined by the convolution kernel fk . The next step involves modeling the three resultant BRPI histograms using the WCGSM model described in [21]. The WCGSM distribution, characterized by two parameters, is given by

WCGSM φ, λ ≡ p brpi 1 − λ2 χ arccos (χ)

= 0.5 + 1 , 2π 1 − χ 2 1 − χ2 (10) where

χ = λ cos brpi − φ + π .

(11)

The parameters of the WCGSM model are estimated as discussed in the following. First, it was empirically observed that the location parameter φ can be accurately estimated using the circular mean of the BRPI, which is defined as n

1 exp j brpi,i , (12) φˆ = arg n i=1 1

Processing the polar representation of the phase chip rather than the phase chip itself allows for easy accounting for phase wrapping. This idea is borrowed from the field of directional (also known as circular) statistics (see, for example, [25–27]). The term “circular” stems from the fact that the phase values are on a circle where the angles π and −π meet at the same point. In directional statistics, phase wrapping is also called phase circularity. This is not to be confused with the circularity and noncircularity discussed later in this paper.


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where n is the total number of samples in the BRPI. The parameter λ is the shape parameter of the WCGSM distribution, which is adaptively estimated based on a simple fitting procedure utilizing the Jensen–Shannon (JS) divergence (see [21] for details). Once the three histograms pertaining to the three kernels are modeled using the three WCGSM models, features based on the WCGSM probability density function (pdf) can be extracted (see Section VII-B-1). Besides the features based on the WCGSM model, it is possible to extract useful features directly from the BRPI given in (9). One such important measure is circularity or noncircularity, which is also known as propriety or impropriety [28, 29]. Circularity means that the BRPI has a pdf that is invariant under rotation in the complex plane. This also implies that the BRPI is uncorrelated with its complex conjugate. In [21, 22] it was shown that using the BRPI, it is possible to characterize the noncircularity in the phase chip. This is achieved by using the modulus of the pseudocovariance [28, 30, 31]: || = E BRPI BRPIT , 0 ≤ || < 1, (13) where E{·} is the expectation, BRPI is the polar representation of BRPI after conversion to a 1-D vector, and superscript T denotes the transpose. Note that if || = 0, then BRPI is said to be second-order circular, or proper. Further, the angle of is used in this study as ∠ = arg {} .

(14)

IV. LINEAR TRANSFORMATION OF 2-D SAR CHIPS TO 1-D SPACE

This section presents three complementary algorithms to transform the 2-D SAR chip into an abstract 1-D vector that accounts for the pixel neighborhood. Our method is inspired by the Radon transform. The remainder of this section is organized as follows: First, a description of the forward Radon transform is given in Section IV-A. Second, a method for linear transformation of the real and the imaginary parts of the complex-valued 2-D SAR chip into a 1-D vector is presented in Section IV-B. Third, a method for linear transformation of the bivariate 2-D SAR chip into a 1-D vector is described in Section IV-C. Finally, Section IV-D describes a method for linear transformation of the complex-valued 2-D SAR chip into a 1-D vector. A. The Forward Radon Transform

The Radon transform Rθ (x ) for a 2-D function f(x, y) is the line integral of f parallel to the y axis, defined as [32]

Rθ x =

∞

f x cos θ − y sin θ, x sin θ + y cos θ dy ,

−∞

(15) where θ is the projection angle and (x , y ) are the projection coordinates, which are related to the projection 790

Fig. 3. Illustration of Radon transform for projection angle θ. Random shape provided represents 2-D function f(x, y).

Fig. 4. Proposed procedure for transforming real and imaginary parts of 2-D complex-valued SAR chip into 1-D vector.

angle by

x y

cos θ = − sin θ

sin θ cos θ

x . y

(16)

The geometry of the Radon transform is illustrated in Fig. 3. Note that the (x , y ) coordinate is rotated about the center of the image as shown in Fig. 3. Among the main advantages of the Radon transform are that it is computed directly in the spatial domain and it is a linear transform [32]. Hence, it preserves the statistics present in the original 2-D SAR chip without introducing any nonlinear artifacts. B. A Method for Linear Transformation of the Real and the Imaginary Parts of the 2-D SAR Chip Into a 1-D Vector

In this subsection, a procedure is proposed for transforming the real and imaginary parts of the 2-D complex-valued SAR chip into a 1-D vector utilizing the Radon transform (Fig. 4). The forward Radon transform is applied separately to the real and imaginary parts of the complex-valued SAR chip (5):

I θ, x = Rθ {i (x, y)}|θ=[0,π ) (17)

Q θ, x = Rθ {q (x, y)}|θ=[0,π ) .

(18)

Note that angles in the range [π, 2π] are omitted, because their corresponding Radon transform provides identical values to angles in the range [0, π). This redundant information is of no interest in this study. Also, note that the Radon transform representation given by (17) and (18) is known as a sinogram. In the next step, the projection angles θ = [0, π) are integrated-out. This is achieved


Fig. 5. Proposed procedure for transforming bivariate 2-D SAR chip into 1-D vector.

through applying the Radon transform to the sinograms at a projection angle φ = π/2:

I x = Rφ I θ, x φ= π (19) 2

Q x = Rφ Q θ, x φ= π .

(20)

2

The output given by (19) and (20) is an abstract 1-D vector representative of the real and imaginary parts of the complex-valued SAR chip. C. A Method for Linear Transformation of the Bivariate 2-D SAR Chip Into a 1-D Vector

In this subsection, a procedure is proposed for transforming the bivariate SAR chip into a real-valued 1-D vector (Fig. 5). The term “bivariate” is used here to denote that the real and imaginary parts of the complex-valued SAR chip are treated as two separate real-valued chips. This is in analogy to the bivariate distribution (e.g., bivariate Gaussian), which is used to model complex-valued data in such a manner (see [29], p. 20). The procedure proposed here is meant to account for the bivariate interactions between the real and imaginary parts of the complex-valued SAR chip. The two sinograms outputted from (17) and (18) are combined together into a single sinogram as follows:

IQ θ, x = I θ, x Q θ, x . (21) Note that MATLAB notation is used in (21) to denote that the two sinograms are concatenated horizontally, along the second dimension. Thus, the resultant sinogram has the same number of rows as in the original sinogram—that is, I (θ, x ) and Q(θ, x ) have similar dimensions—and the number of columns is doubled. In the next step, the projection angles θ = [0, π) are integrated out. This is achieved through applying the Radon transform to the combined sinogram output from (21) at a projection angle φ = π/2 as follows:

IQ x = Rφ IQ θ, x φ= π . (22) 2

The output given by (22) is an abstract 1-D vector representative of the bivariate statistics in the input 2-D complex-valued SAR chip.

Fig. 6. Proposed furuding procedure.

D. A Method for Linear Transformation of the Complex-Valued 2-D SAR Chip Into a 1-D Vector

The procedure described in Section IV-C accounts for the bivariate statistics between the real and imaginary parts of the complex-valued SAR chip. However, the complex-valued statistics [29] are not meant to be accounted for by this procedure. Hence, to account for the composite interactions within and between the real and imaginary parts, a simple procedure is proposed in this subsection. First, the real and the imaginary parts of the complex-valued SAR chip are suitably amalgamated in the spatial domain as fuiq (x, y) = furud (i (x, y) , q (x, y)) .

(23)

This specific form of interleaving is referred to as furuding, inspired by the spectroscopic binary in the constellation Canis Major known by the traditional name Furud [33, 34]. Fig. 6 demonstrates our proposed furuding procedure. In the next step, the real-valued furud chip is transformed to a 1-D vector by inputting it to the proposed algorithm introduced in Fig. 4. The final output is given by

Fuiq x = Rφ Rθ fuiq (x, y) θ=[0,π ) π . (24) φ= 2

V. ADAPTIVE AND COMPLEXITY MEASURES BASED SOLELY ON 1-D REPRESENTATIONS

In this section, the adaptive and complexity measures considered in this study and based on the 1-D representation of the SAR chip are presented. Some of these measures are directly based on the 1-D Radon representations discussed earlier, while some others are based on the Hilbert spectrum computed from the 1-D Radon representations. The remainder of this section is


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organized as follows: In Section V-A, a brief overview of the Hilbert–Huang transform [12] is presented, along with a few proposed modifications. Further, in Section V-B, a selection of relevant methods used in this study for quantifying the nonlinear dynamics is provided. A. Hilbert–Huang Transform (HHT)

The HHT represents an advancement in nonlinear and nonstationary signal processing [12]. First, it uses a technique known as empirical mode decomposition to decompose the data, according to their characteristic scales, into a set of intrinsic mode functions (IMFs). Thus, unlike Fourier-based methods, the basis of the data comes from the data themselves. Second, the IMFs are used to construct a time/space-frequency-energy distribution known as the Hilbert spectrum. Subsequently, the time/space localities of the events are preserved. Therefore, the frequency and energy defined by the Hilbert transform have intrinsic and instantaneous physical meaning. Although the terms “spectrum” and “frequency” are traditionally associated with Fourier-based analysis, the HHT provides a different interpretation for these terms. In doing so, the HHT avoids the Heisenberg principle, which is a serious setback to all Fourier-based time/space-frequency methods including the Fourier-based wavelet transform [12, 35].2 In this study, ensemble empirical mode decomposition (EEMD) [36] is applied to the outputs given by (19) and (20) separately. EEMD is a noise-assisted method which resolves the problem of mode mixing encountered in traditional empirical mode decomposition [12]. Primarily, EEMD decomposes the input data to a small number of IMFs based on the local characteristic time/space scale. An IMF represents a simple oscillatory mode as a counterpart to the harmonic function. By definition, an IMF is any function with the same number of extrema and zero-crossings, with its envelopes being symmetric with respect to zero. This definition guarantees a well-behaved Hilbert transform of the IMF. The procedure of extracting an IMF is called sifting. In our subsequent analysis, we use a local stopping criterion for the sifting process as prescribed in [36]. Thus, m IMFs are extracted from I (x ) and Q(x ) as

I x = IIMF x a + rI x m

a=1

QIMF x a + rQ x , Q x = m

(25)

a=1

2

To clarify, the HHT does not, of course, overcome the Heisenberg principle. Rather, it avoids the issue through adaptively extracting the bases of the data to be analyzed from the data themselves. In doing so, the HHT does not impose any foreign bases on the data, as is the case with Fourier-based methods. Further, because the bases are adaptively extracted, the HHT removes the necessity, associated with Fourier-based methods, of having to deal with the unavoidable compromise in presetting time/space and frequency resolutions of the bases.

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where rI (x ) and rQ (x ) are the residues. The total number of IMFs is specified as [12]

m = log2 n − 1, (26) where n is the length of the original 1-D signal. In order to calculate the Hilbert spectrum, two methods are used in this study [37, 38]: direct quadrature (DQ) and generalized zero-crossing (GZC). The DQ method is based on the analytic signal for each IMF. Traditionally, the HHT achieves this through computing the Hilbert transform for the IMF and then placing the result in the imaginary part of the analytic signal. The real part is the IMF signal itself. However, according to the Paley–Wiener theorem, the complex-valued signal output from the quadrature demodulator is indeed an analytic signal, whose imaginary part is simply the Hilbert transform of its real part [39]. Hence, we form the analytic signals based on the proper combination of the real and imaginary parts for each IMF according to

HDQ x a = IIMF x a + j QIMF x a , (27) where a ∈ {1, . . . , m}. Note that this representation is known as the Hilbert spectrum. Thus, the instantaneous magnitude of the Hilbert spectrum is given by

2

2 0.5 . (28) bDQ x a = IIMF x a + QIMF x a Further, the unwrapped instantaneous phase for the Hilbert spectrum is computed as

hDQ x a = unwrap arg HDQ x a , (29) where unwrap denotes the addition of multiples of ± 2π when absolute jumps between consecutive elements of [hDQ (x )]a are greater than or equal to the default jump tolerance of π radians. Subsequently, the instantaneous frequency is computed, following the definition of the principle of stationary phase, as

d IFDQ x a = (30) hDQ x a . dx In the GZC method, the instantaneous frequency [IFGZC (x )]a is defined for the whole wave (i.e., based on the real part of each IMF in our case), which includes the values from crest to crest, trough to trough, and up (down) zero-crossing to up (down) zero-crossing. The corresponding bGZC (·) is denoted [bGZC (x )]a (see [37, 38] for details). B. Quantifying the Nonlinear Dynamics

In this subsection, the measures for nonlinearity used in this study are presented. First, the order of nonlinearity is described in Section V-B-1. Second, the degree of nonlinearity is elaborated on in Section V-B-2. Third, the combined degree of nonlinearity is described in Section V-B-3. Fourth, the permutation entropy is introduced in Section V-B-4. Finally, the scaling exponent, based on detrended fluctuation analysis, is discussed in Section V-B-5.


1) Order of Nonlinearity: The average frequency for the cycles of the instantaneous frequency and the real part of each IMF, respectively, are computed from

(31) [fIF ]a = favg IFDQ x a

[fI ]a = favg IIMF x a .

TABLE I Illustration of Subset Encoding and Rank Ordering

Index

Encoding Symbol

Signal Sequence

Rank-Ordered Sequence

Encoded Sequence

1 2 3

αβγ αβγ αβγ

3, −100, 0 −100, 0, 2 0, 2, 5

−100, 0, 3 −100, 0, 2 0, 2, 5

βγ α αβγ αβγ

(32)

Note that [fIF ]a is the intrawave frequency and [fI ]a is the corresponding oscillation frequency. Thus, we compute the nonlinear order for each IMF as [12, 37, 40] [O]a =

[fIF ]a + 1. [fI ]a

(33)

The inequality [fIF ]a /[fI ]a < 1 implies that [fIF ]a is undersampled. This means that the estimate of [O]a based on such values is incorrect or at least inaccurate. Thus, we do not use the undersampled frequencies for order estimation in this study. Note that in (31) and (32) it is possible to replace the average with the median [41]. 2) Degree of Nonlinearity: For each IMF, the degree of nonlinearity DNa can be computed as [37]

bGZC x a IFDQ x a − IFGZC x a , DNa = std [IFGZC (x )]a bGZC (x ) a

(34) where std denotes the standard deviation and [bGZC (x )]a is the mean value of [bGZC (x )]a . It is possible to replace std with the interquartile range (IQR) [42]. It is also possible to replace the mean value of [bGZC (x )]a with the median. 3) Combined Degree of Nonlinearity: The combined degree of nonlinearity CDN weighs the degree of nonlinearity for each IMF by the energy in each IMF as [37]

2 m IIMF x a CDN = DNa m , (35) 2 k=1 [IIMF (x )]k a=1 where [IIMF (x )]2a is the energy in the real part of the analytic signal pertaining to each IMF. 4) Permutation Entropy: Permutation entropy (PE) [43, 44] is a simple yet robust complexity measure for a time series based on its neighboring values. In analogy to relevant measures for chaotic dynamic systems, PE behaves similarly to the Lyapunov exponent, and it is found to be useful particularly in the presence of nonlinear dynamics in the signal. To take into account the causal information pertaining to any effects stemming from the temporal order of the successive elements of the time series, the time series is encoded first into sequences of symbols, based on the theory of symbolic dynamics. Then the entropy is computed for the encoded sequence as follows [43]: PE = −

l! c=1

p` c log2 p` c .

(36)

TABLE II Relative Frequencies of All Possible Patterns in x(n) as Illustrated in Table I c

Pattern

1 2 3 4 5 6

αβγ αγ β βαγ βγ α γ αβ γ βα

Relative Frequency (p` c ) 1 3

+

1 3

=

2 3

0 0 1 3

0 0

Here, p` c represents the relative frequencies of the possible patterns of symbol sequences, termed permutations, and l is an important parameter for the number of possible permutation patterns. Also, note that in computing p` c there is an important parameter called τ which describes the time delay between successive points in the symbol sequence. Detailed description for the PE algorithm as well as important practical recommendations for choosing the two parameters can be found in [45]. To illustrate the utility of PE, a simple example follows. Consider the signal to be analyzed, x (n) = {3, −100, 0, 2, 5} .

(37)

Assume l = 3 and τ = 1. Given these parameters, the process of computing PE involves encoding the data sequence into subsets of three symbols (i.e., l = 3) with a time lag of 1 (i.e., τ = 1) between the successive subsets. Then the encoded subsets are rank ordered. The overall process is shown in Table I. Because l = 3, the overall number of possible permutations is 3! = 6. Hence, six patterns are tracked in the encoded sequence, and the relative frequency of each pattern (i.e., p` c ) is computed as shown in Table II. Accordingly, PE can be computed by substituting the values for p` c found in Table II into (36), as 2 1 2 1 log PE = − + log2 = 0.9183. (38) 3 2 3 3 3

5) Scaling Exponent Based on Detrended Fluctuation Analysis: Detrended fluctuation analysis is a simple technique for identifying the extent of fractal self-similarity in a nonstationary time series based on the calculation of a scaling exponent α. First, the time series ˜ to be analyzed is integrated to produce a self-similar x(n)


793

TABLE III List of the MSTAR Targets Used in This Study

random walk [46, 47] n˜ ˜ = x ρ . y (n)

(39)

ρ=1

Target Name

No. of Training Chips (17◦ Depression Angle)

No. of Testing Chips (15◦ Depression Angle)

˜ is successively subdivided into windows of Then y(n) length L samples. For a time series of length M samples, there will be scales equal to the nearest integer to log2 M. A least-squares straight-line local trend is calculated by analytically minimizing the squared error E2 over the slope and intercept parameters a and b as [46, 47]

BTR-60 2S1 BRDM-2 D7 T62 ZIL-131 ZSU-23/4 SLICY

256 299 298 299 299 299 299 298

195 274 274 274 273 274 274 274

arg min E = 2

a,b

L

˜ − a n˜ − b)2 . (y (n)

(40)

˜ n=1

Then the fluctuation is calculated over all windows at each time scale as [46, 47] 0.5 L 1 2 ˜ − a n˜ − b) (y (n) F (L) = . (41) L n=1 ˜ On a log-log graph of L versus F(L), a straight line indicates self-similarity expressed as F (L) ∝ α. The scaling exponent α is calculated as the slope of a straight line fitted to the log-log graph of L versus F(L) using least-squares as before (see [46, 47] for more details). VI. THE SAR DATA SET

This study utilizes a comprehensive public-domain single-channel (i.e., HH polarization) and single-look complex-valued SAR data set collected and distributed under the DARPA Moving and Stationary Target Acquisition and Recognition (MSTAR) program [48]. Sandia National Laboratories used an X-band STARLOS sensor in spotlight mode to collect the data. The MSTAR data set provides a nominal spatial resolution of 0.3047 × 0.3047 m in both range and azimuth. The data used in this study come from two CDs available from the Sensor Data Management System title MSTAR/IU Mixed Targets CD1 and CD2. In total, for each CD there are eight different types of stationary target imaged at azimuth angles covering the full span of [0◦ , 360◦ ). CD1 and CD2 include SAR data collected at 15◦ and 17◦ depression angles, respectively. In this paper, the 17◦ data set is used for training the classifier, while the 15◦ data set is used for testing the classifier. A list of the target names and the overall number of the complex-valued SAR chips used in this study is provided in Table III. Ground-truth pictures for the eight targets are depicted in Fig. 7. VII. FEATURE EXTRACTION

This section presents the three feature sets utilized in this study. First, the baseline features extracted from the power-detected SAR chips are provided in Section VII-A. Then the holism-based features, extracted from both the phase chips and the 1-D representations, are introduced in Section VII-B. Finally, feature normalization is described in Section VII-C. 794

Fig. 7. Ground-truth pictures for MSTAR targets used in this study [48].

A. Baseline Features

Baseline features are solely based on the power-detected SAR chip. These features represent the common practice of discarding the phase in the single-channel SAR literature, and they are used for comparison purposes to demonstrate the uniqueness and independence of the information carried in the holism-based features. Nineteen baseline features are utilized in this study. The procedure for extracting the baseline features is summarized in Fig. 8. First, the


Fig. 8. Procedure for extraction of baseline features.

complex-valued SAR chip is power detected as p (x, y) = [i (x, y)]2 + [q (x, y)]2 .

(42)

Then the power-detected SAR chip is thresholded through an adaptive information-theoretic approach based on the entropy of the histogram, as originally proposed by Kapur et al. [49]. This method was chosen because it is found to offer excellent performance. Further, morphological dilation is applied to the thresholded image. This operation is aimed at merging the relevant different connected regions in the thresholded image into one contiguous region representative of the target extent. In the next step, a set of features is extracted from the binary image, the dilated binary image, and the gray-level image. These extractions are meant to represent the power-based (i.e., intensity) features commonly used in the literature, and they include the following [50]: • Number of scattering centers (fBL1 ): the number of connected regions in the binary image. • Area (fBL2 ): the total number of pixels with value of 1 in the binary image. • Centroid (fBL3 , fBL4 ): the center of mass of the dilated binary image. Note that the first element (fBL3 ) is the horizontal coordinate (or x-coordinate) of the center of mass and the second element (fBL4 ) is the vertical coordinate (or y-coordinate). • Major-axis length (fBL5 ): the length (in pixels) of the major axis of the ellipse that has the same normalized second central moments as the region. This measure is based on the dilated binary image. • Minor-axis length (fBL6 ): the length (in pixels) of the minor axis of the ellipse that has the same normalized second central moments as the region. This measure is also based on the dilated binary image. • Eccentricity (fBL7 ): the eccentricity of the ellipse that has the same second moments as the region. The eccentricity is the ratio of the distance between the foci of the ellipse and its major axis length. The value is between 0 and 1. This measure is also based on the dilated binary image. • Orientation (fBL8 ): the angle (in degrees, ranging from −90◦ to 90◦ ) between the x-axis and the major axis of the ellipse that has the same second moments as the region. This measure is also based on the dilated binary image. • Convex area (fBL9 ): the number of pixels in the convex hull that specifies the smallest convex polygon that can contain the region. This measure is also based on the dilated binary image.

• Euler number (fBL10 ): the number of objects in the region minus the number of holes in those objects. This measure is based on the binary image. • Equivalent diameter (fBL11 ): the diameter of a circle with √ the same area as the region. Computed as (4/π )fBL2 . This measure is based on the dilated binary image. • Solidity (fBL12 ): the proportion of the pixels in the convex hull that are also in the region, computed as fBL2 /fBL9 . This measure is also based on the dilated binary image. • Extent (fBL13 ): the ratio of pixels in the region to pixels in the total bounding box. Computed as fBL2 /(area of the bounding box). This measure is also based on the dilated binary image. • Perimeter (fBL14 ): the distance between each adjoining pair of pixels around the border of the region. This measure is also based on the dilated binary image. • Weighted centroid (fBL15 , fBL16 ): the center of the region based on location and intensity value. The first element (fBL15 ) is the horizontal coordinate (or x-coordinate) of the weighted centroid. The second element (fBL16 ) is the vertical coordinate (or y-coordinate). This measure is based on both the dilated binary image and the power-detected intensity image. • Mean intensity (fBL17 ): the mean of all the intensity values in the region of the power-detected image as defined by the dilated binary image. • Minimum intensity (fBL18 ): the value of the pixel with the lowest intensity in the region of the power-detected image as defined by the dilated binary image. • Maximum intensity (fBL19 ): the value of the pixel with the greatest intensity in the region of the power-detected image as defined by the dilated binary image.

B. Holism-Based Features

In this study, there are two sets of holism-based features. First, the adaptive statistical features based solely on the phase chip are presented in Section VII-B-1. Then the adaptive and complexity features based solely on the 1-D representations of the SAR chip are described in Section VII-B-2. It may be noted that because the sideband responses due to nonlinear scattering are distributed over an area far larger than that physically occupied by the target [16], no segmentation should be utilized, and the largest area possible around the target should be included. Hence, the entire MSTAR chip provided by the Sensor Data Management System is used for extracting the holism-based features. 1) Adaptive Statistical Features Based Solely on the Phase Chip: While various types of features can be extracted based on the BRPI as well as the WCGSM model described in Section III, a set of 15 features are considered in this study for demonstration purposes:


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• The location parameter based on the first kernel (fPh1 ): the location parameter φˆ of the WCGSM pdf for the first kernel (see Fig. 2), estimated based on (12). • The shape parameter based on the first kernel (fPh2 ): the shape parameter λˆ of the WCGSM pdf for the first kernel (see Fig. 2), estimated based on the JS divergence method (see [21]). • Maximum peak value for the first kernel (fPh3 ): the peak value of the WCGSM pdf, based on the first kernel. • The location parameter based on the second kernel (fPh4 ): the location parameter φˆ of the WCGSM pdf for the second kernel (see Fig. 2), estimated based on (12). • The shape parameter based on the second kernel (fPh5 ): the shape parameter λˆ of the WCGSM pdf for the second kernel (see Fig. 2), estimated based on the JS divergence method (see [21]). • Maximum peak value for the second kernel (fPh6 ): the peak value of the WCGSM pdf, based on the second kernel. • The location parameter based on the third kernel (fPh7 ): the location parameter φˆ of the WCGSM pdf for the third kernel (see Fig. 2), estimated based on (12). • The shape parameter based on the third kernel (fPh8 ): the shape parameter λˆ of the WCGSM pdf for the third kernel (see Fig. 2), estimated based on the JS divergence method (see [21]). • Maximum peak value for the third kernel (fPh9 ): the peak value of the WCGSM pdf, based on the third kernel. • First psuedocovariance measure (fPh10 ): the measure given in (14), based on the first kernel. • Second pseudocovariance measure (fPh11 ): the measure given in (14), based on the second kernel. • Third pseudocovariance measure (fPh12 ): the measure given in (14), based on the third kernel. • First noncircularity measure (fPh13 ): the measure given in (13), based on the first kernel. • Second noncircularity measure (fPh14 ): the measure given in (13), based on the second kernel. • Third noncircularity measure (fPh15 ): the measure given in (13), based on the third kernel.

2) Adaptive and Complexity Features Based Solely on 1-D Representations: Prior to converting the SAR chips from 2-D to 1-D space (see Section IV), it is important that all the SAR chips are zero-padded to a standardized size. This guarantees the compatibility of similar feature measures extracted from different target chips. For the MSTAR data set considered, it is noted that the size of the SAR chips varies from 54 × 54 pixels (for SLICY) to 193 × 192 pixels (for ZIL-131). Hence, each SAR chip is zero-padded on all sides to yield a standardized size of 200 × 200 pixels. Then the adaptive and complexity measures presented in Section V are invoked. In total, 98 features are extracted from the 1-D representations of the SAR chip: • Features based on the IMFs (f1D1 tof1D32 ): The 1-D representation of the real and imaginary parts of the complex-valued SAR chip is decomposed using EEMD into eight distinct IMFs pairs. For each pair of IMFs, the 796

average frequency is calculated based on the real part (f1D1 ) as described by (32), the average instantaneous frequency based on the real and imaginary parts (f1D2 ) as described by (31), the order of nonlinearity (f1D3 ) as estimated by (33), and the degree of nonlinearity (f1D4 ) as shown in (34). • Combined degree of nonlinearity (f1D33 ): calculated based on (35). • Features based on the PE for the combination of IMF pairs (f1D34 tof1D41 ): The real and imaginary parts of each IMF pair are combined into one vector ([[IIMF (x )]a , [QIMF (x )]a ]), and then the PE is computed for the combined vector as described in Section V-B-4. Note that all the PE computations in this paper are based on order of appearance, where first occurrence implies a lower rank (see [45] for details). The number of permutation patterns l is set to 3. The time delay τ between successive points in the symbol sequence is set to 1. These parameters are used in all subsequent computations of the PE, and they were chosen following the recommendations provided in [45]. • Features based on the PE of the instantaneous frequency of the Hilbert spectrum calculated based on the DQ method (f1D42 tof1D49 ): the PEs of the eight vectors [IFDQ (x )]a produced by (30). • Features based on the PE of the magnitude of the Hilbert spectrum calculated based on the DQ method (f1D50 tof1D57 ): the PEs of the eight vectors [bDQ (x )]a produced by (28). • Features based on the PE of the instantaneous frequency of the Hilbert spectrum calculated based on the GZC method (f1D58 tof1D65 ): the PEs of the eight vectors produced from [IFGZC (x )]a (see Section V-A). • Features based on the PE of the magnitude of the Hilbert spectrum calculated based on the GZC method (f1D66 tof1D73 ): the PEs of the eight vectors produced from [bGZC (x )]a (see Section V-A). • Features based on the PE of the instantaneous frequency of the Hilbert spectrum calculated based on the combination of the DQ and GZC methods (f1D74 tof1D81 ): the PEs of the eight vectors produced from the combination given by [[IFDQ (x )]a , [IFGZC (x )]a ]. • Features based on the PE of the magnitude of the Hilbert spectrum calculated based on the combination of the DQ and GZC methods (f1D82 tof1D89 ): the PEs of the eight vectors produced from the combination given by [[bDQ (x )]a , [bGZC (x )]a ]. • Features based on the PE of the 1-D Radon signals (f1D90 tof1D94 ): a set of features directly extracted from the 1-D Radon representation for the real part (f1D90 ) described in (19), the imaginary part (f1D91 ) described in (20), the combination of the real and imaginary parts into a 1-D vector (f1D92 ), the bivariate representation (f1D93 ) described in (22), and the furuded representation (f1D94 ) described in (24). • Fluctuation-index features based on the 1-D Radon signals (f1D95 tof1D98 ): a set of features directly extracted from the 1-D Radon representation for the real part


(f1D95 ), the imaginary part (f1D96 ), the bivariate representation (f1D97 ), and the furuded representation (f1D98 ). C. Feature Normalization

Feature standardization and feature scaling are two important aspects pertaining to feature normalization in this study. In the machine-learning literature (e.g., [51]) and its applications to target detection and classification in SAR imagery (e.g., [52, 53]), feature standardization makes the values of each feature in the data set have zero mean and unit variance and is defined as f − μˆ fZ = , (43) σˆ where f is feature vector, μˆ is the sample mean of f, and σˆ is the sample standard deviation of f. In statistics, this standardization procedure is known as the standard score or the z-score [54]. Major assumptions in [43] are that the data in f follow the Gaussian distribution and that the sample μˆ and σˆ are similar or at least close to the population’s mean μ and standard deviation σ . The second assumption is often unrealistic. For the feature set considered in this study, we noted that the feature vectors are not Gaussian distributed, and we found that standardization following [43] degrades the classification accuracy of the classifier. Accordingly, μˆ and σˆ , respectively, are replaced with the median and the IQR as follows: f − median fR = . (44) IQR These two measures are borrowed from the field of robust statistics [41]. Robust statistics seeks to provide methods that emulate popular statistical methods but are not unduly affected by outliers or other small departures from model assumptions. The median, being the numerical value separating the higher and lower halves of a data sample, is a robust measure of central tendency, while μˆ is not. The IQR is a measure of statistical dispersion, being equal to the difference between the upper and lower quartiles; it is the most significant basic robust measure of scale. If there are outliers in the data, then the IQR is more representative than σˆ as an estimate of the spread of the body of the data [42]. Once the feature vectors are standardized, the next step involves feature scaling. Feature scaling is an important step that prevents attributes in greater numeric ranges from dominating those in smaller numeric ranges. Following the recommendations in [55], each training feature vector is scaled first in the range [−1, 1]; then the corresponding testing feature vector is scaled based on the minimum and maximum values in the training feature vector, not the testing vector.

this study, and a brief description of the classifier design procedure. In Section VIII-B, a method for feature ranking and selection is presented. A. Classifier Design

Here, the LIBSVM software system [56] is used to design multiclass support vector machine (SVM) classifiers. SVMs are a powerful supervised classification technique that takes advantage of the so-called kernel trick [14, 51]. The main idea of an SVM is that the feature data are mapped to a much higher dimension than the original space. In the high-dimensional space, data from two classes can always be linearly separated by a hyperplane. After the linear decision boundary is determined, the data are then projected back to the original dimension of the feature space. This procedure is motivated by Cover’s theorem, which states, “a complex pattern classification problem, cast in a high dimensional space nonlinearly, is more likely to be linearly separable than in a low dimensional space, provided that the space is not densely populated” [57]. Based on the training data, the SVM produces a model that allows prediction of the target values of the test data given only the test-data attributes. The building block of the multiclass SVM is a binary classifier (also known as a dichotomizer). The multiclass classifier can be composed based on the one-against-one approach. For a number of classes K, the total number of dichotomizers needed is given by [51] Number of dichotomizers =

K (K − 1) . 2

(45)

For example, in the SAR data set considered in this study there are eight target classes; hence, 28 dichotomizers are required. Each dichotomizer should be trained on the combination of two classes. Thus, for a particular dichotomizer, given a training set of instance-label pairs (xq , yq ), where q = {1, . . . , l}, xq ∈ Rn , and yq ∈ {ωi , ωj }l , the SVM requires solving the following optimization problem [55, 56, 58, 59]: min

Wωi ωj ,bωi ωj ,ξ ωi ωj

1 ωi ωj T ωi ωj ωi ωj ) (W )+C (ξ ) , (W 2 q

subject to

(Wωi ωj )T φ xq + bωi ωj ≥ 1 − (ξ ωi ωj )q if xq ∈ ωi ,

(Wωi ωj )T φ xq + bωi ωj ≤ −1 + (ξ ωi ωj )q if xq ∈ ωj , ξi ≥ 0.

(46)

VIII. CLASSIFIER DESIGN AND FEATURE SELECTION

Here, C > 0 is the penalty or regularization parameter for the error term and φ(·) is a higher dimensional space function that defines the kernel function as

This section is composed of two parts: Section VIII-A contains a description of the classification method used in

K xi , xj ≡ φ(xi )T φ xj .


(47) 797

The kernel used in this work is the Gaussian radial basis function, defined as [51, 55]

2 , (48) K xi , xj = exp −γ xi − xj where γ > 0 is the kernel parameter. This kernel was chosen because it is found to give excellent performance for our feature set. The two parameters (C, γ ) play a crucial role in dictating the performance of the SVM classifier. Following the guidelines in [55, 56, 60], we adopt a grid-search and a ν-fold cross validation to find the optimal values of these parameters. For each dichotomizer, given a testing instance xtest , the decision function (predictor) is [56] g ωi ωj (xtest ) = sgn (Wωi ωj )T φ (xtest ) + bωi ωj . (49) In the classification stage, a voting strategy is deployed based on the votes cast by each dichotomizer for all data points xtest . Hence, a point with the maximum number of votes is designated to be in the class [56, 60, 61]. B. Feature Ranking and Selection

In this study, a method for feature ranking is used to evaluate the information content in each feature vector. Further, the best subset of features are selected for classifier construction. The strategy utilized is adopted from [62, 63]. Primarily, a combination of the Fisher score (F-score) and SVM multiclass classification is used. The F-score is a technique for measuring the information contained in a feature vector based on its between-classes and within-class discriminability. Given a feature vector Xi with K classes (i.e., k = 1, . . ., K), denote the set of 2 instances in class k as X(k) i and define the L -norm (k) (k) lk = |Xi |. Further, assume x¯ i and x¯ i are the averages of the ith feature in X(k) i and Xi , respectively. The F-score of the ith multiclass feature vector is defined as [51, 62–64], F (i) =

SB (i) , SW (i)

(50)

with SB (i) =

K 2 ¯ lk x¯ (k) − x i i

(51)

k=1

SW (i) =

K

2 xi − x¯ (k) . i

(52)

k=1 xi ∈ Xi(k)

The numerator indicates the discrimination between the classes, and the denominator indicates the discrimination within the class. The larger the F-score, the more likely the feature is discriminative. Once the F-score is computed for each feature vector, the features are sorted based on their significance as defined by the F-score. Then the high-F-score features are added gradually and used to train the multiclass SVM classifier. This process is continued until the validation accuracy of the classifier decreases. The subset of features that achieves the highest validation accuracy is selected [62, 64]. 798

IX. RESULTS

Using the training and testing data sets presented in Section VI, three different training and testing feature sets were extracted. The first feature set contained the baseline features based on the power-detected SAR chips described in Section VII-A. In total, there were 19 baseline features for each target chip. Hence, the size of the constructed baseline features matrix for training was 2347 × 19 and for testing was 2112 × 19. The second feature set was for the phase-based features presented in Section VII-B-1. Fifteen phase-based features were extracted from each target chip. The size of the phase-based features matrix for training was 2347 × 15 and for testing was 2347 × 15. The third feature set was solely based on the 1-D representations, introduced in Section V. A total of 98 features were extracted from each target chip. Accordingly, the size of the 1-D-based features matrix for training was 2347 × 98 and for testing was 2112 × 98. Each feature set was normalized following the procedure prescribed in Section VII-C. Then the classification accuracy of each feature set was investigated. Additionally, the F-score was computed for each feature vector. This captures the information content in each feature vector separately, and independent of other features. Further, the combinatorial effect on the classification accuracy was examined for the combinations of baseline and phase-based features, holism-based (i.e., phase-based and 1-D-based) features, and baseline and holism-based features. This served to explore the interindependence between the three feature sets. Following the steps outlined in Section VIII-A, three different multiclass SVM classifiers were trained, using the baseline, phase-based, and 1-D-based training feature sets. The grid search for the optimal values of (C, γ ) pertaining to the three classifiers is depicted in Fig. 9. The accuracy depicted is based on a fivefold cross validation. Optimal values were found to be (25 , 2−1 ), (2, 2), and (215 , 2−11 ), respectively, for the baseline, phase-based, and 1-D-based features. Once the three classifiers were constructed based on the optimal parameters found, they were tested using the corresponding testing feature sets. The confusion matrices for the baseline and the holism-based (i.e., phase-based and 1-D-based) classifiers are provided in Figs. 10–12, respectively. The arrangement of the targets in the confusion matrices follows Table III. The overall classification accuracy (as well as the validation accuracy), calculated based on [56], is Accuracy =

No. of correctly predicted data × 100. (53) No. of testing data

Now the classification results for the holism-based features are explained and compared with those for the baseline features. First, the classification results obtained for the phase-based features (see Fig. 11) evidently demonstrate that the phase in single-channel SAR imagery is not useless, as is often assumed in the literature. On the contrary, based on only 15 features an overall classification accuracy of 63.02% was achieved. It is


Fig. 9. Grid search for optimal (C, γ ) for three feature sets.

Fig. 10. Confusion matrix for baseline classifier. Classification accuracy = 73.63% (1555/2112).

interesting to note that for four targets (BTR-60, 2S1, ZSU-23/4, and SLICY), higher classification accuracy was achieved in comparison to the baseline features (see Fig. 10). This implies that these targets possess greater effects due to dispersive or nonlinear scattering manifested in their corresponding phase images. For the classification result of the 1-D-based features given in Fig. 12, it is noted that targets SLICY, ZIL-131, D7, 2S1, ZSU-3/4, and T62 achieved higher classification accuracy, while targets BTR-60 and BRDM-2 scored relatively lower classification accuracy, all with reference to the baseline features (see Fig. 10). This indicates that the 1-D-based features provide for improvement in the overall classification accuracy; the interindependence between the

three feature sets warrants investigation, which is addressed later in this section. Because sideband responses are resolution dependent, it should be noted that if a SAR data set with higher resolution is used, one would expect an additional increase in the classification accuracy for the holism-based features. Next, the features were ranked using the F-score, and a search for the best subset of features within each feature set was conducted as outlined in Section VIII-B. Recall that the F-score is computed for each feature separately, hence this measure is for the information content in each feature independent of the number of features. The F-scores pertaining to features in baseline, phase-based, and 1-D-based sets, respectively, are depicted in


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Fig. 11. Confusion matrix for phase-based classifier. Classification accuracy = 63.02% (1331/2112).

Fig. 12. Confusion matrix for 1-D-based classifier. Classification accuracy = 79.12% (1671/2112).

Fig. 13. Significance of baseline features. Feature index represents feature subscripts provided in Section VII-A.

Figs. 13–15. To visually demonstrate the F-score measures of the features relevant to each other, the three F-score figures are combined into Fig. 16. The F-score values are given in Tables IV–VI, respectively. Generally, the F-score results convey that the significance of some phase-based features (fPh3 ) for discrimination between the target classes is around 7 times that of the baseline features (fBL5 ). Furthermore, the significance of some 1-D-based features (f1D91 ) is around 160 times that of the baseline features (fBL5 ). This shows the utility of the holism-based approach. For each of the three sets of features, a search for the best subset led to the conclusion that all the features are important in attaining the classification accuracy achieved. Next, combinatorial effects pertaining to the three sets of features were studied. Three more classifiers were trained and tested following the procedure outlined earlier. 800

Fig. 14. Significance of phase-based features. Feature index represents feature subscripts provided in Section VII-B-1.

The first classifier was based on the amalgamation of the baseline and the phase-based features. Thus, the size of the new feature matrices for training and testing, respectively, were 2347 × (19 + 15) and 2112 × (19 + 15). The second classifier used the amalgamation of the phase-based and 1-D-based features; hence, the size of the features matrix for training was 2347 × (15 + 98) and for testing was 2112 × (15 + 98). The third classifier utilized the combination of the baseline, phase-based, and 1-D-based features. Accordingly, the size of the feature matrix for training was 2347 × (19 + 15 + 98) and for testing was 2112 × (19 + 15 + 98). The optimal values for (C, γ ) were obtained based on a grid search (see Fig. 17) and found to be (25 , 2−1 ), (29 , 2−9 ), and (25 , 2−5 ) for the first, second, and third classifiers, respectively. The confusion matrices and classification accuracy for the three classifiers are provided in Figs. 18–20, respectively.


Fig. 15. Significance of 1-D-based features. Feature index represents feature subscripts provided in Section VII-B-2.

Fig. 16. Significance of all features (baseline, phase-based, and 1-D-based) used in this study. Blue, red, and green bars, respectively, represent baseline, phase-based, and 1-D-based features.

TABLE IV F-Scores for the Baseline Features Used in This Study Index Type F-score

1 (64) fBL1 0.104282

2 (59) fBL2 0.174448

3 fBL3 0.026250

4 (51) fBL4 0.240112

5 (30) fBL5 0.525163

6 (63) fBL6 0.116269

7 (39) fBL7 0.356272

8 fBL8 0.007372

Index Type F-score

9 (52) fBL9 0.234009

10 fBL10 0.074707

11 (50) fBL11 0.252705

12 fBL12 0.071361

13 fBL13 0.090576

14 (36) fBL14 0.435705

15 fBL15 0.037509

16 (53) fBL16 0.205777

Index Type F-score

17 (42) fBL17 0.296468

18 fBL18 0.037128

19 (44) fBL19 0.290143

Note: Bold numbers in parentheses indicate the ranks for the top 66 features.


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TABLE V F-Scores for the Phase-Based Features Used in This Study Index Type F-score

20 fPh1 0.002161

21 (15) fPh2 3.40389

22 (13) fPh3 3.93829

23 fPh4 0.001028

24 (21) fPh5 0.852989

25 (22) fPh6 0.850224

26 fPh7 0.000536

Index Type F-score

28 (35) fPh9 0.437565

29 fPh10 0.003383

30 fPh11 0.004468

31 fPh12 0.001335

32 (14) fPh13 3.609093

33 (32) fPh14 0.466928

34 (66) fPh15 0.100274

27 (34) fPh8 0.451283

Note: Bold numbers in brackets indicate the ranks for the top 66 features.

TABLE VI F-Scores for the 1-D-Based Features Used in This Study Index Type F-score

35 f1D1 0.004376

36 f1D2 0.043474

37 f1D3 0.028188

38 (23) f1D4 0.837737

39 (16) f1D5 2.561608

40 f1D6 0.083659

41 (17) f1D7 1.675789

42 (49) f1D8 0.255196

Index Type F-score

43 (19) f1D9 1.151379

44 (24) f1D10 0.831957

45 f1D11 0.048254

46 (38) f1D12 0.383011

47 (37) f1D13 0.402271

48 (28) f1D14 0.569529

49 f1D15 0.003632

50 (55) f1D16 0.202775

Index Type F-score

51 f1D17 0.038605

52 f1D18 0.043167

53 f1D19 0.028673

54 (65) f1D20 0.101209

55 f1D21 0.068022

56 f1D22 0.032381

57 f1D23 0.005276

58 f1D24 0.044145

Index Type F-score

59 f1D25 0.006465

60 f1D26 0.035941

61 f1D27 0.025960

62 f1D28 0.061196

63 f1D29 0.011671

64 f1D30 0.026343

65 f1D31 0.004708

66 f1D32 0.015550

Index Type F-score

67 (20) f1D33 0.937748

68 (5) f1D34 41.99154

69 (18) f1D35 1.335222

70 (25) f1D36 0.737384

71 (47) f1D37 0.258012

72 f1D38 0.021904

73 f1D39 0.020498

74 f1D40 0.008744

Index Type F-score

75 f1D41 0.005032

76 (2) f1D42 74.732794

77 f1D43 0.012247

78 (27) f1D44 0.661262

79 (31) f1D45 0.521751

80 f1D46 0.06181

81 f1D47 0.094429

82 f1D48 0.021047

Index Type F-score

83 f1D49 0.057071

84 (4) f1D50 45.77489

85 f1D51 0.045835

86 (26) f1D52 0.698165

87 (48) f1D53 0.257581

88 (58) f1D54 0.191538

89 (54) f1D55 0.205569

90 f1D56 0.035003

Index Type F-score

91 f1D57 0.031306

92 (12) f1D58 5.651479

93 f1D59 0.092150

94 (62) f1D60 0.127972

95 f1D61 0.026360

96 f1D62 0.056588

97 f1D63 0.087552

98 f1D64 0.009154

Index Type F-score

99 f1D65 0.011370

100 (11) f1D66 5.932869

101 (29) f1D67 0.55627

102 (40) f1D68 0.348003

103 (61) f1D69 0.143246

104 f1D70 0.035961

105 f1D71 0.058544

106 f1D72 0.003314

Index Type F-score

107 f1D73 0.005743

108 (3) f1D74 49.33849

109 f1D75 0.029424

110 (46) f1D76 0.261102

111 (56) f1D77 0.202594

112 f1D78 0.041158

113 f1D79 0.067312

114 f1D80 0.003950

Index Type F-score

115 f1D81 0.004851

116 (10) f1D82 34.94347

117 f1D83 0.035165

118 (33) f1D84 0.452377

119 (60) f1D85 0.173644

120 f1D86 0.090287

121 f1D87 0.067904

122 f1D88 0.003530

Index Type F-score

123 f1D89 0.006756

124 (6) f1D90 41.93384

125 (7) f1D91 40.41706

126 (1) f1D92 84.95294

127 (9) f1D93 37.15177

128 (8) f1D94 38.85961

129 (43) f1D95 0.296341

130 (41) f1D96 0.318477

Index Type F-score

131 (57) f1D97 0.196293

132 (45) f1D98 0.261724

Note: Bold numbers in parentheses indicate the ranks for the top 66 features.

802


Fig. 17. Grid search of optimal (C, γ ) for combinations of three feature sets.

Fig. 18. Confusion matrix for classifier based on baseline and phase-based features. Classification accuracy = 81.39% (1719/2112).

Fig. 19. Confusion matrix for classifier based on phase-based and 1-D-based features. Classification accuracy = 93.42% (1973/2112). EL-DARYMLI ET AL.: HOLISM FOR TARGET CLASSIFICATION IN SYNTHETIC APERTURE RADAR IMAGERY

803

Fig. 20. Confusion matrix for classifier based on baseline, phase-based, and 1-D-based features. Classification accuracy = 96.35% (2035/2112).

Fig. 21. Grid search for optimal (C, γ ) pertinent to selected 66 features.

These results demonstrate the interindependence between the two sets of the holism-based (i.e., phase-based and 1-D-based) features relative to the baseline features. Particularly, when the baseline and phase-based features were combined, the classification accuracy increased by around 8%. Furthermore, combination of the phase-based and 1-D-based features allowed for an overall classification accuracy of 93.42%. This significant improvement in the classification accuracy shows the importance of these two sets of features. Additionally, inclusion of the baseline features slightly increased the classification accuracy to 96.35%. Next, a search for the best subset of features from the combination of the three sets was conducted. First, the features were sorted based on the F-score values provided in Tables IV–VI (see also Fig. 16). Further feature subset selection was conducted following the procedure outlined in Section VIII-B. This led to choosing the top 66 features, as they were found to achieve the highest validation accuracy (96.17%). Ranks for the selected 66 features are shown between brackets in Tables IV–VI. It is worth noting that among the 66 features selected, 54 come from the holism-based features, of which 29 are the top ranked. Finally, the selected features were used to construct a new SVM classifier. A grid search was conducted (see Fig. 21) and the optimal values of (C, γ ) were found to be (25 , 2−5 ). Once the classifier was constructed, it was tested using the corresponding 66 features selected from the combined testing data set. The confusion matrix for this 804

classifier is provided in Fig. 22. Based on this result, it is clear that the classification accuracy for the classifier based on the selected 66 features was very close to that for the classifier based on the whole set of 132 features. The most prominent lessons learned from this investigation are now highlighted. First, contrary to the usual practice of discarding the phase in single-channel SAR imagery under the assumption that it carries no useful information, the statistical significance of the information carried in the phase in general, and the complex-valued chip in particular, was clearly demonstrated here. Second, a classification accuracy of 93.42% was achieved for the holism-based (i.e., phase-based and 1-D-based) features. This validates the performance of the proposed framework both for phase characterization and modeling and for transformation from 2-D to 1-D space. This also demonstrates that by using the correct set of features, it is possible to neglect features based on the detected SAR chips. Hence, through approaching the complex-valued SAR chip from the holism-based perspective proposed in this paper, it is possible to gain a new insight into the process of feature extraction for target recognition in SAR imagery. Third, it is interesting to note that among the most significant features were those based on the PE for the combination of the 1-D transformed real and imaginary parts, real part, imaginary part, and furuded and bivariate representations, respectively. This validates the usefulness of the three Radon-based 1-D representations proposed in this study. This also demonstrates the prominence of PE at capturing nonlinear dynamics in the different representations. Additionally, among the top-ranked features were those based on the Hilbert spectrum. This shows the superiority of the HHT at capturing nonlinear dynamics. Next, it is important to note that the feature-extraction methods presented in this study are by no means exhaustive. Rather, they serve to demonstrate the objective of the study and they open the door for more in-depth investigation into various adaptive, complex-valued, nonlinear, and nonstationary feature-extraction methods. Finally, it should be stressed that with the increase in the spatial resolution of the SAR sensor relative to the size of the imaged target, the nonlinear phenomenon due to dispersive scattering is naturally expected to be more pronounced. Thus, the application of our approach to this kind of SAR imagery should achieve even more prominent


Fig. 22. Confusion matrix for classifier based on selected 66 features. Classification accuracy = 96.07% (2029/2112).

classification accuracy. Future work may consider investigating additional holism-based methods for signal analysis, besides the effect of different classification methods on classification accuracy.

ACKNOWLEDGMENT

The authors would like to thank Dr. Paris Vachon from Defence R&D Canada (Ottawa), who provided advice and oversight to the project team for the last five years. REFERENCES

X. CONCLUSIONS

For the case of extended targets and due to dispersive scattering, phase change in the radar return signal is not linear, as is often assumed by conventional resolution theory. In fact, nonlinear phase modulation due to dispersive scattering is an intrinsic phenomenon for extended targets in SAR imagery. When the complex-valued image output from SAR processors is approached from the reductionist perspective of linear signal processing, the effect of nonlinearity is often viewed as a noise that warrants removal. On the contrary, because holism-based methods for signal analysis account for multiplicity due to interactions between the individual components in the signal, this paper has clearly demonstrated the advantage of such methods for target-recognition applications in SAR imagery. Particularly, two holism-based frameworks for feature extraction from complex-valued SAR imagery have been presented. The first framework is based solely on the phase chip, while the second utilizes 1-D representations for the complex-valued SAR chip. Using the real-world MSTAR data set, representative features under each framework are presented and compared with baseline features extracted from the power-detected SAR chip. An overall improvement in the classification accuracy of around 20% is achieved due to the proposed approach. Further, using the F-score to measure the information content in each feature vector independently, top-ranked features from the first and second holism-based frameworks, respectively, are found to be 7 and 160 times that of the baseline features. This shows the superiority of the holism-based approach. The higher the spatial resolution of the SAR sensor relative the size of the imaged target, the more predominant the dispersive scattering in the processed complex-valued image. Hence, the proposed approach is expected to offer even greater gains for such sensors. The application of our approach extends well beyond SAR to include various kinds of relevant signals, such as that from coherent radar generally, sonar, and ultrasound.

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Khalid El-Darymli (M’08) received a B.Sc. degree in electrical engineering from the Garyounis University of Libya, an M.Sc. degree in computer and information engineering from International Islamic University Malaysia, and a Ph.D. degree in electrical engineering (defense was passed with distinction) from the Memorial University of Newfoundland, St. John’s, Newfoundland, Canada. He is currently a senior engineer with Northern Radar, Inc. His special research interests include holism, nonlinear and dispersive scattering, adaptive, nonlinear and chaos-theory-inspired methods for signal processing, target recognition in radar imagery, and software-defined radio. He is a member of the Association of Professional Engineers and Geoscientists of Alberta and a fellow of the School of Graduate Studies at the Memorial University of Newfoundland. EL-DARYMLI ET AL.: HOLISM FOR TARGET CLASSIFICATION IN SYNTHETIC APERTURE RADAR IMAGERY

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Peter McGuire received B.Asc. (Eng.) and Ph.D. degrees in aerospace engineering from the University of Toronto. He studied the use of artificial neural networks for computer vision and control of dynamic systems at the University of Toronto. Since joining C-CORE, he has specialized in image processing, earth observation, and data-fusion projects. His projects include earth observation using a virtual synthetic aperture radar constellation of satellites, sensor management, and data-fusion system design using holonic control, along with high-speed automated inspection using computer vision. In addition to project-related activities, he is cross-appointed at the Memorial University of Newfoundland, where he manages a research program focused on oil- and gas-related issues. His topics include detection and mapping of oil under ice using autonomous underwater vehicles, advanced techniques for monitoring targets and infrastructure using satellite and ground-based synthetic aperture radar, coordination of aerial and ground-based robotic systems, and sense-and-avoid algorithms for unmanned aerial vehicles. Eric W. Gill (M’00—SM’05) received a B.Sc. degree (1st class) in physics from the Memorial University of Newfoundland, St. John’s, Newfoundland, Canada, in 1977 and M.Eng. and Ph.D. degrees in electrical engineering from the same institution in 1990 and 1999, respectively. For over two decades, starting in 1977, he was a lecturer in physics and mathematics in the provincial college system, and for a significant period during those years he pursued research interests in rough-surface electromagnetic scattering theory. He is currently a professor in the department of electrical and computer engineering at Memorial University of Newfoundland, where he teaches and conducts research in theoretical and applied electromagnetics. His special interest lies in the scattering of high-frequency electromagnetic radiation from time-varying, randomly rough surfaces, with particular application to the use of high-frequency surface-wave radar in remote sensing of the marine environment. His latest pursuits include ocean remote sensing using both X-band nautical radar and synthetic aperture radar. He is a member of the American Geophysical Union and a senior member of the IEEE. Desmond Power (M’11) has received M.Eng. and B.Eng. degrees. He started his career in terrestrial radar, working as an RF designer in over-the-horizon radar. He was involved in signal processing and analysis of radar data. After the launch of RADARSAT in 1995, he moved into projects related to earth observation, with his first project dealing with iceberg detection capabilities of synthetic aperture radar. Since that time, he has managed and been a technical advisor to a large series of projects with C-CORE involving earth observation, including marine target detection, vehicle detection along rights-of-way, and interferometry for ground deformation measurement. He has over 24 years of experience in radar and remote sensing. He is currently the vice president of remote sensing with C-CORE. He is actively involved in the development of terrestrial-based radar systems. He is a principal investigator of the multimillion-dollar research and development project on radar-based critical infrastructure monitoring funded by the Atlantic Innovation Fund. He is a member of the Association of Professional Engineers and Geoscientists of Newfoundland and Labrador.

Cecilia Moloney (M’91) received a B.Sc. (Hons.) degree in mathematics from the Memorial University of Newfoundland, Canada, and M.Asc. and Ph.D. degrees in systems-design engineering from the University of Waterloo, Canada. From 2004 to 2009, she held the NSERC/Petro-Canada Chair for Women in Science and Engineering, Atlantic Region. Since 1990 she has been a faculty member with the Memorial University of Newfoundland, where she is currently a full professor of electrical and computer engineering. Her research interests include nonlinear signal- and image-processing methods, signal representations via wavelet and contourlet transforms, radar signal processing, transformative pedagogy for science and engineering, and gender and science studies. 808


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