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Sparse Signal Methods for 3D Radar Imaging Christian D. Austin, Emre Ertin, and Randolph L. Moses The Ohio State University, Department of Electrical and Computer Engineering 2015 Neil Avenue, Columbus, OH 43210, USA Email: {austinc, ertine, randy}@ece.osu.edu

Abstract—Synthetic aperture radar (SAR) imaging is a valuable tool in a number of defense surveillance and monitoring applications. There is increasing interest in three-dimensional (3D) reconstruction of objects from radar measurements. Traditional 3D SAR image formation requires data collection over a densely sampled azimuth-elevation sector. In practice, such a dense measurement set is difficult or impossible to obtain, and effective 3D reconstructions using sparse measurements are sought. This paper presents wide-angle three-dimensional image reconstruction approaches for object reconstruction that exploit reconstruction sparsity in the signal domain to ameliorate the limitations of sparse measurements. Two methods are presented; first, we use ℓp penalized (for p ≤ 1) least squares inversion, and second, we utilize tomographic SAR processing to derive wideangle 3D reconstruction algorithms that are computationally attractive but apply to a specific class of sparse aperture samplings. All approaches rely on high-frequency radar backscatter phenomenology so that sparse signal representations align with physical radar scattering properties of the objects of interest. We present full 360◦ 3D SAR visualizations of objects from air-toground X-band radar measurements using different flight paths to illustrate and compare the two approaches.

I. I NTRODUCTION There is increasing interest in three-dimensional (3D) reconstruction of objects from radar measurements. This interest is enabled by new data collection capabilities, in which airborne synthetic aperture radar (SAR) systems are able to interrogate a scene, such as a city, persistently and over a large range of aspect angles [1]. Three-dimensional reconstruction is further motivated by an increasingly difficult class of surveillance and security challenges, including object detection and activity monitoring in urban scenes. Additional information provided by wide-aspect 3D reconstructions can be useful in applications such as automatic target recognition (ATR) and tomographic mapping. In SAR imaging, an aircraft emits electromagnetic signal pulses along a flight path and collects the returned echoes. The returned echoes can be interpreted as one-dimensional lines of the 3D Fourier transform of the scene, and the aggregation of radar returns over the flight path defines a manifold of data in the scene’s 3D Fourier domain [2]. A number of techniques have been proposed for narrow angle 3D reconstruction from This material is based upon work supported by the Air Force Office of Scientific Research under Award No. FA9550-06-1-0324. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the Air Force. C. Austin was supported in part by a fellowship from the Ohio Space Grant Consortium. This paper is an extension of work previously presented in [30]–[32].

this manifold of data. Two different approaches to 3D image formation are full 3D reconstruction and 2D non-parametric imaging followed by parametric estimation of the third, height dimension. Full 3D reconstruction methods invert an operator to retrieve the three-dimensional reflectivity function; specifically, in SAR imaging, the operator can be modeled as a Fourier operator, since data is collected over a manifold in 3D Fourier space of the scene. Generating high-resolution 3D images using traditional Fourier processing methods requires that radar data be collected over a densely sampled set of points in both azimuth and elevation angle, for example, by collecting data from many closely spaced linear flight passes over a scene [3], [4]. This method of imaging requires very large collection times and storage requirements and may be prohibitively costly in practice. There is thus motivation to consider more sparsely sampled data collection strategies, where only a small fraction of the data required to perform traditional high-resolution imaging is collected. Sparsely sampled data collections with elevation diversity can be achieved through nonlinear flight paths [5]–[8]. However, when inverse Fourier imaging is applied to sparsely sampled apertures, reconstruction quality can be poor. Reconstruction quality can be quantified by the point spread function (PSF) of the image, defined by the Fourier transform of the data aperture indicator function. The mainlobe of this PSF will typically be wider (indicating reduced resolution) and the sidelobes higher than for the PSF of a reconstruction formed from a densely-sampled measurement aperture (see e.g. [7], [9]). Methods to mitigate this problem by deconvolving the PSF from the 3D reflectivity function using greedy algorithms were investigated in [5], [6]. In this paper we develop an ℓp regularized least squares approach to wide-angle 3D radar reconstruction for arbitrary, sparse apertures, and we demonstrate this approach on the problem of 3D vehicle reconstruction. A second approach is based on forming a small set of 2D SAR images followed by parametric 1D estimation to estimate the third, or height, dimension in the backscatter profile. We refer to these as 2D+1D techniques. Interferometric SAR (IFSAR) is a well-known classical technique for parametric height estimation from 2D SAR imagery formed at two linear elevation passes [2], [10]; in this image formation method, only a sparse data collection in elevation is required to form 3D images. More recently, multi-baseline extensions, sometimes referred to as Tomographic SAR or Tomo-SAR, have been developed for height estimation (see, e.g. [11]–[20]). In TomoSAR imaging, height estimation can be treated as a sinusoids-

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in-noise problem and can be solved using spectral estimation techniques. A number of different techniques exist for estimating the height-dependent reflectivity of a scene, including the RELAX algorithm [11], uniform and non-uniform Fourier beamforming [12], [13], [16], Truncated SVD processing [14], Capon filtering [16], parameter estimation methods [18], regularized optimization algorithms [19], [20], and the ESPRIT algorithm [21]; a survey of different height estimation methods is given in [17]. Differential techniques also exist [19], [22], in which passes are collected at different times and temporal velocity is also estimated. These techniques can be formulated as 2D+2D approaches. Many of these approaches have been applied to strip-map SAR processing of large scenes, and 3D reconstruction is generally aimed at reconstruction of buildings, forest canopies, or nonuniform terrain height. Furthermore, the approaches have been applied to relatively narrow angle collection geometries, in which an isotropic scattering assumption is well-approximated. In both 2D+1D and full 3D reconstructions, radar scattering is typically anisotropic over wide angles and violates the isotropic point scattering assumption of traditional radar imaging. As a result, reconstructed image resolution will be worse than indicated by PSF analysis, which is predicated on an isotropic scattering assumption [23]. In this paper, we develop and compare two techniques, one a full 3D method, and one 2D+1D based, for achieving accurate 3D scene reconstructions from sparse, wide-angle measurement apertures. Both techniques rely on some basic properties of scattering physics, and both exploit signal sparsity (in the reconstruction domain) of radar scenes. In particular, we are interested in imaging man-made structures under high-frequency radar operation. Under these operating conditions, scenes are dominated by a sparse number of dominant isolated scattering centers; dominant returns result from objects such as corner or plate reflectors made from electromagnetic conductive material (see e.g. [24]). The first algorithm, ℓp regularized least-squares (LS) processing, is a full 3D approach, requires only knowledge of the flight geometry, and is applicable to “image formation” in arbitrary collection geometries. In this paper we will use the term “image formation” to denote both 2D and 3D radar scene reconstructions. In addition, since collected radar data can be interpreted as samples in the 3D Fourier transform space of the scene, matrix-vector multiplications in the regularized LS algorithm can be replaced by the Fast Fourier Transform (FFT). The regularized LS approach is also known as Basis Pursuit Denoising when p = 1 [25], [26]. This approach has been shown to produce well-resolved, 2D SAR image reconstructions over approximately linear flight paths [27]– [29] it was also used for 3D image reconstruction in [30]– [32]. This ℓp regularized LS approach is also used in TomoSAR to resolve scatterers in the height dimension as an alternative to spectral estimation [19]. For wide-angle 3D SAR reconstruction, a direct implementation of ℓp regularized LS methods yields a prohibitively large optimization problem; one of the contributions of this paper is the development of a computationally tractable implementation. The second reconstruction algorithm we consider is a Tomo-SAR-based

approach adapted to address vehicle-sized 3D reconstruction over very wide azimuth data collections, including full 360◦ circular SAR; this approach exploits knowledge of scattering sparsity to improve height resolution [32], [33] and is computationally faster than the first algorithm; however, it applies to a more restrictive class of sparse data collection schemes. Anisotropic scattering over wide angles is addressed in both algorithms by using non-coherent subaperture imaging, where scattering is assumed to be isotropic over narrow-angle subapertures. We investigate the two wide-angle 3D SAR image formation methods on different sparse data collection geometries. The first collection geometry is a pseudorandom path collection for three polarizations generated by Visual-D electromagnetic simulation software, and released as a public dataset by the Air Force Research Laboratory (AFRL) [34]. The second dataset, also released by AFRL, is from an actual 2006 multipass X-band Circular SAR (CSAR) data collection of a ground scene [35]. This dataset consists of eight fully circular paths in azimuth, at eight closely-spaced elevation angles with respect to scene center; this data is polarimetric, in that horizontalhorizontal (HH), vertical-vertical (VV) and cross-polarization data is collected. We generate pseudorandom dataset images using the regularized LS algorithm and multipass CSAR dataset images using both algorithms; the Tomo-SAR approach requires multipass data and cannot be applied to the pseudorandom path data. The previously-discussed resolution and sidelobe issues that result from sparse measurement apertures are manifest in both of these datasets. The contributions of this paper can be summarized as follows. First, we propose a technique to process sparse wideangle data, such as circular SAR data, for object reconstruction; this type of data is becoming increasing important in persistent surveillance applications. Second we provide full 3D radar reconstructions using ℓp regularized sparsity techniques, and provide a tractable algorithm, in terms of memory and computational requirements, for generating full 3D reconstructions from arbitrary, sparse 3D flight paths. Third, we demonstrate the first high-fidelity 3D vehicle reconstructions from an arbitrary curvilinear flight path. Fourth, we successfully demonstrate multi-baseline tomographic SAR for 3D reconstructions of passenger vehicles from airborne measurements using full 360◦ azimuth data from an operational 0.3m resolution X-band radar. Finally, we provide an initial comparison of Tomo-SAR and ℓp regularized LS approaches on both synthetic and measured X-band radar data of vehicles, in terms of both reconstruction performance and computational cost. An outline of the paper is as follows. First, an overview of the SAR data model is presented in Section II. Section III describes the two collection geometries, pseudorandom and CSAR, and corresponding datasets considered here. These collection geometries demonstrate some of the challenges presented by such sparse collections. In Section IV, the ℓp regularized LS imaging algorithm is presented, and in Section V the wide-angle Tomo-SAR algorithm is discussed. Section VI presents reconstructed 3D images of vehicles from both the pseudorandom and CSAR data collection geometries. Finally,



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Section VII concludes, summarizing the main results of the paper. II. SAR M ODEL In this section, we briefly review the tomographic SAR model used for reconstruction. We assume that the radar transmits a wideband signal with bandwidth BW centered about a center frequency fc . Such a signal could be an FM chirp signal or a stepped-frequency signal, but other wideband signals can also be used. We also assume that the transmitter is sufficiently far away from the scene so that wavefront curvature is negligible, and we use a plane wave model for reconstruction; this assumption is valid, for example, when the extent of the scene being imaged is much smaller than the standoff distance from the scene to the radar. For a radar located at azimuth φ and elevation θ with respect to scene center that transmits an interrogating signal, the received waveform, in the far-field case, is given by [2] r(t; φ, θ, pol) = # "Z Bz˜ Z By˜ (1) ct y d˜ z ⋆ s(t), g x ˜ = , y˜, z˜; φ, θ, pol d˜ 2 −Bz˜ −By˜ where c is the speed of light, s(t) is a known, bandlimited signal with center frequency fc and bandwidth BW that represents the transmitted waveform convolved with antenna responses; pol is the polarization of the transmit/receive signal pair, and ⋆ denotes convolution. The x ˜-coordinate is defined as the radial line from the radar to scene center, and y˜, and z˜ are orthogonal to x ˜ and to each other. This coordinate system is a translation in x ˜ from scene center and a rotation by (φ, θ) of a fixed, ground coordinate system (x, y, z), whose origin is at scene center. The scene’s reflectivity function is given by g(˜ x, y˜, z˜; φ, θ, pol), or equivalently, by g(x, y, z; φ, θ, pol) in a fixed ground coordinate system. Boundaries of the scene in each dimension are denoted as B(·) . Under the far-field assumption, these boundaries are assumed to be sufficiently small so that waveform curvature and range-dependent signal attenuation can be neglected, which means that these scene boundaries are on the order of objects but not entire large scenes. For large scenes, (1) applies locally around setpoints of interest. Equation (1) can be interpreted as the Fourier transform of the scene reflectivity function projected onto the x ˜-dimension. By the projection-slice theorem [2], this Fourier transform is equivalent to a line along the x ˜-axis in 3D spatial frequency space, or k-space, of the scene reflectivity function. Specifically, the 3-D Fourier transform G(kx , ky , kz ) of the reflectivity function g(x, y, z; φ, θ, pol), observed from angle (φ, θ) at polarization pol is given by: Z G(kx , ky , kz ; φ, θ, pol) = g(x, y, z; φ, θ, pol) (2) e−j(kx x+ky y+kz z) dx dy dz. The frequency support of each measurement is a line rad/m centered at segment in (kx , ky , kz ) with extent 4πBW c 4πfc c rad/m, and oriented at angle (φ, θ). The flight path defines which line-segments in k-space are collected, and hence what

subset of k-space is sampled. Typically both the frequency variable f along each line segment and the flight path are sampled as f → fj , (φ, θ) → (φn , θn ), so one obtains a set of k-space samples indexed on (j, n) as: kxj,n

=

kyj,n

=

kzj,n

=

4πfj cos θn cos φn c 4πfj cos θn sin φn c 4πfj sin θn . c

(3)

In order to use tomographic inversion techniques to recover g from k-space measurements, it is often assumed in (2) that the scene reflectivity is isotropic; so, g(x, y, z; θ, φ, pol) is not a function of θ and φ. For narrow-angle measurements, this assumption is generally valid; however, for wide-angle measurements, the isotropic scattering assumption is not valid for most scattering centers in the scene [36], [37]. One approach for reconstruction from wide-angle measurements, and the one adopted in this paper, is to subdivide the measurements into a set of possibly overlapping subapertures and to assume scattering is locally isotropic on each subaperture. Once subaperture reconstructions are obtained, one can then form an overall wide-angle reconstruction by combining the narrow-aperture reconstructions in an appropriate way. In particular, we argue that a good way to implement the subaperture combination is using a Generalized Likelihood Ratio Test (GLRT) approach. We assume scattering at each point (x, y, z) in the scene can be characterized by a limitedangle response centered at azimuth φ and elevation θ and with some persistence width in each angular dimension. We treat the persistence angle as fixed and known, and we use this to establish the angular widths of the subapertures used in the data formation. Since the response center angles (φ, θ) are unknown, we estimate them using a GLRT formulation: use a bank of matched filters, each characterized by a center response azimuth and a response width and shape, and compute the response amplitude as I(x, y, z; pol) = arg max |I(x, y, z; φ, θ, pol)|, (φ,θ)

(4)

where I denotes the matched filter output. The maximization in equation (4) over continuous-valued φ and θ is approximated by discretizing these two variables. Since backprojection radar image formation can be interpreted as a matched filter for point scattering responses [38], each matched filter output I(x, y, z; φ, θ, pol) is well-approximated by the subaperture radar image formed from k-space measurements at discrete center angles (φj , θj ) and with fixed azimuth and elevation extent. That is, the approach of forming subaperture radar reconstructions, then combining these reconstructions by taking the maximum over all subapertures, can be interpreted as a GLRT approach to reconstruction of limited-persistence scattering centers. While the approach in (4) assumes that all scattering centers have identical and known persistence, generalizations to variable persistence angles can also be developed [39]. As a side note, each voxel in the image reconstruction is also characterized by the maximizing (φ, θ)



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dataset consists of k-space samples computed along a continuous, far-field pseudo-random “squiggle” path in azimuth and elevation from a construction backhoe vehicle. The path is intended to simulate an airborne platform that interrogates the object over a wide range of azimuth and elevation angles, but doing so while flying along a 1D curved path that sparsely covers the 2D azimuth-elevation angular sector. Three polarizations are included in the dataset, vertical-vertical (VV), horizontal-horizontal (HH), and cross-polarization (HV). The trace in Figure 1 shows the path as a function of azimuth and elevation angle, defined with respect to a fixed ground plane coordinate system, and Figure 2(a) displays the corresponding k-space data that can be collected by the radar, which is contained between the inner and outer domes that denote the minimum and maximum radar frequency, respectively. The squiggle path is superimposed on the outer dome. The set of k-space data collected along the squiggle path is very sparse with respect to the full data dome. The azimuth and elevation extents of the squiggle path are approximately [66◦ , 114.1◦ ], and [18◦ , 42.1◦ ], respectively. This range of nearly 50◦ in azimuth and 25◦ in elevation indeed represents wide-angle measurement at X-band; the persistence of many scattering centers at X-band has been reported to be (significantly) lower [36], [37]. In contrast, a filled aperture used to form benchmark 1 ◦ in this azimuth/elevation images uses samples at every 14 sector. 45 40 35 Elevation

center angles, providing additional information useful for image visualization [29] or object recognition [40], [41]. In the algorithms presented below, we will assume that the available k-space data are partitioned into (possibly overlapping) subapertures, and that reconstructions for each subaperture are obtained using the proposed algorithms. Then, a final wide-angle reconstruction is obtained using (4). An advantage of this locally-isotropic approach is that, for each subaperture, scattering responses are parameterized by only location and amplitude. An alternate approach, considered in [24], [42]–[44], is to adopt models that directly characterize anisotropic scattering. One can then directly estimate scattering centers from the entire wide-angle data using these models. This latter approach may be posed as a classical parametric model order and parameter estimation problem (see, e.g., [24], [44]). Alternately, one can adopt a nonparametric approach in which anisotropic scattering is characterized as a linear combination of dictionary elements that are limited in persistence, and one estimates the amplitudes of a sparse linear combination of dictionary elements; such an approach has recently been proposed in [42]. A related nonparametric approach is to estimate an image at each aspect angle from a sparse linear combination of dictionary elements; the images are not independently formed, but linked through a regularization term that penalizes for large changes in pixel magnitudes that are close in aspect [43]; regularization enforcing sparsity in these nonparametric approaches is similar to the ℓp reconstruction technique presented in Section IV below. These wide-angle nonparametric approaches result in a (much) larger set of dictionary elements than used in the approach followed here; this is because anisotropic scattering is characterized by additional parameters such as orientation and persistence angles. In principle, the approaches in [24], [42]–[44] are based on similar assumptions, but represent different algorithmic approaches to estimate the reconstruction. A detailed comparison of these approaches in terms of both computation and reconstruction performance remains a topic for future study.

30 25 20 15

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III. C OLLECTION G EOMETRY AND E XAMPLE DATASETS Before presenting the proposed reconstruction approaches, it is useful to examine some example data collection apertures and the associated 3D reconstruction challenges that result. We will first present and discuss two sparse radar collection geometries and their associated reconstruction objectives. The first dataset considered is synthetically generated data from a pseudorandom flight path developed by researchers at AFRL as a 3D image reconstruction challenge problem [34]; the second dataset is a collection of X-band field measurements from a CSAR radar at eight closely-spaced elevations [35]. A. Pseudorandom Flight Path Dataset The pseudorandom flight path dataset [34], [45] is generated by the Visual-D electromagnetic scattering simulator. The simulator models scattering returns from a radar with center frequency fc = 10 GHz and bandwidth BW = 6 GHz. The

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85

90 95 Azimuth

100

105

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Fig. 1. Sparse “squiggle”path radar measurements as a function of azimuth and elevation angle in degrees.

B. Multipass Circular SAR Dataset The second sparse dataset we consider is the multipass CSAR data from the AFRL GOTCHA Volumetric SAR Data Set, Version 1.0 [35], [46]. This dataset consists of sampled, dechirped radar return values that have been transformed to the form of G(kx , ky , kz ; φ, θ, pol) in (2). The data is fully polarimetric from 8, 360◦ CSAR passes. The planned nominal collection consists of passes at constant equallyspaced elevation angles with respect to scene center, with elevation difference, ∆el = 0.18◦ , in the range [43.7◦ , 45◦ ]. The actual flight path is not perfectly circular, as shown in Figure 3, and not at perfectly constant and equally-spaced elevations. The center frequency of the radar is fc = 9.6GHz,



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IV. ℓp R EGULARIZED L EAST-S QUARES I MAGING A LGORITHM

(a) Squiggle Path

In this section we present the first of two 3D imaging algorithms; this algorithm applies to general data collection scenarios, but will be used for sparse collections here. The proposed approach assumes that the number of 3D locations in which nonzero backscattering occurs is sparse in the 3D reconstruction space, and applies sparse reconstruction techniques. We pose the reconstruction as an ℓp regularized least-squares (LS) problem, in which a regularizing term encourages sparse solutions. This ℓp regularized LS imaging algorithm attempts to fit an image-domain scattering model to the measured kspace data under a penalty on the number of non-zero voxels. The algorithm assumes that the complex magnitude response of each scattering center is approximately constant over narrow aspect angles and across the radar frequency bandwidth. The algorithm in this section applies to general apertures; this is in contrast to the second algorithm presented in Section V, which applies to apertures with specific structure. Define a set of N locations in image reconstruction space as candidate scattering center locations, C = {(xn , yn , zn )}N n=1 .

(b) CSAR Path Fig. 2. Data domes of all k-space data that can be collected by a radar for (a) the pseudorandom synthetic “squiggle” path backhoe dataset, and (b) the GOTCHA dataset; units are in rad/m. Support of the k-space data is contained between the inner and outer dome. Inner and outer domes show the minimum and maximum radar interrogating frequencies. The outlines on the outer domes show the locations of the sparse k-space data collected, which extends from the outline radially to the inner dome.

and the bandwidth of the radar is 640MHz, significantly lower than that of the squiggle path collection. Figure 2(b) shows the k-space data collected by the eight GOTCHA passes. The kspace radial extent from the outer dome to inner dome of data collected, dictated by radar bandwidth, is seen to be significantly smaller than in the squiggle path case. Figure 2(b) also illustrates that the GOTCHA k-space data is very limited in elevation extent, in contrast to the squiggle path.

(5)

Typically these locations are chosen on a uniform rectilinear grid. The M × N data measurement matrix is given by h i A = e−j(kx,m xn +ky,m yn +kz,m zn ) , m,n

where m indexes the M measured k-space frequencies down rows, and n indexes the N coordinates in C across columns. Under the assumption that scattering center amplitude is constant over the aspect angle extent and radar bandwidth considered, the measured (subaperture) data from the scattering center model, (2), can be written in matrix form as b = Aβ + ν,

(6)

where β is the N -dimensional vectorized 3D image that we wish to reconstruct; it has complex amplitude value βn in row n if a scattering center is located at (xn , yn , zn ) and is zero in row n otherwise; the image vector β maps to the 3D image, I(xn , yn , zn ), by the relation I(xn , yn , zn ) = β(i) if and only if column i of A is from coordinate (xn , yn , zn ). The vector ν is an M dimensional i.i.d. circular complex Gaussian noise vector with zero mean and variance σn2 , and b is an M -dimensional vector of noisy k-space radar measurements. ˆ is the solution to the sparse The reconstructed image, β, optimization problem [27], [28] (7) βˆ = argmin kb − Aβk22 + λkβkpp , β

Fig. 3.

Actual GOTCHA passes. Scale is in meters.

where the p-norm is denoted as k · kp , 0 < p ≤ 1, and λ is a sparsity penalty weighting parameter. Note that the solution to (7) applies for general A matrices, and the radar flight path locations that index the rows of A can be arbitrary. In particular, flight paths such as the squiggle path in Figure 2(a) can be used. Many algorithms exist for solving (7) or the constrained version of this problem when p = 1 (e.g. [26], [47]–[50]), or in the more general case, when 0 < p ≤ 1 (e.g.



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[28], [51]). We use the iterative majorization-minimization algorithm in [28] to implement (7). This algorithm is suitable for the general case when 0 < p ≤ 1. The algorithm has two loops, an outer loop which iterates on a surrogate function and an inner loop that solves a matrix inverse using a conjugate gradient algorithm, in our experience, the inner loop terminates after very few iterations when using a Fourier operator, as considered here. Empirical evidence also indicates that this majorization-minimization algorithm terminates faster than a split Bregman iteration approach [50]. An outline of the majorization-minimization algorithm implementation is provided in the Appendix. For algorithm implementation, define a uniform rectilinear grid on the x, y, and z spatial axes with voxel spacings of ∆x, ∆y, and ∆z, respectively. Let the set of candidate coordinates C in (5) consist of all permutations of (x, y, z) coordinates from the partitioned axes; then, the set C defines a uniform 3D grid on the scene. If, in addition, the k-space samples are on a uniform 3D frequency grid centered at the origin, the operation Aβ can be implemented using the computationally efficient 3D Fast Fourier Transform (FFT) operation. In many scenarios, including the one here, the measured k-space samples are not on a uniform grid, and the FFT cannot be used directly. Instead an interpolation step followed by an FFT is needed. An alternative approach would be to use Type-2 nonuniform FFTs (NUFFT)s as the operator A to process data directly on the non-uniform kspace grid, at added computational cost [52], [53]. Nonuniform FFT algorithms require an interpolation step, which is executed each time the operator A is evaluated; whereas, in FFT implementation, interpolation occurs only once and the interpolated data becomes b. When using an iterative algorithm to solve (7), as used here, having to perform interpolation once can result in significant computational savings. Our empirical results on the X-band data sets considered here suggest that nearest neighbor interpolation results in well-resolved images at low computational cost, and so it is adopted here. Implementing the optimization algorithm solving (7) for large-scale problems can be challenging from a memory and computational standpoint. In iterative algorithms, like the one utilized here, typically, the data vector b as well as the current iterate of β and a gradient with the same dimension as β is stored. For example, in the simulations below, we reconstruct a scene with N = 182 × 250 × 252 ≈ 1.1 × 107 voxels to cover a single vehicle. So, at the very least, it would be necessary to store the data vector in addition to two vectors of double or single precision in 1.1 × 107 dimensional complex space. For algorithms that utilize a conjugate gradient approach to calculate matrix inverses, it is also necessary to store a conjugate vector of the same dimension N , and in a Newton-Raphson approach, it is necessary to store a Hessian of dimension N × N . During each iteration of an algorithm, it is commonly required to evaluate the operator A and its adjoint. These operations can become very computationally expensive when the problem size grows and may result in a computationally intractable algorithm, unless a fast operator such as the FFT is employed. Specifically, since A is an M × N matrix, direct multiplication of Aβ requires M N multiplies and additions per

evaluation. In examples using the squiggle path and nine subapertures chosen, the average value of these nine M values is 105 , so M ×N ≈ 1012 operations. After initial interpolation, an FFT implementation of Aβ requires O(D3 log(D3 )) operations, where D is the maximum number of samples across the image dimensions. For the imaging example with dimensions 182 × 250 × 252, D = 252. For concreteness, assuming the constant multiple on the order of operations in the FFT is close to unity, FFT implementation of the operator A requires approximately 2523 log(2523 ) ≈ 3.8×108 operations; so, FFT implementation results in computational savings greater than a factor of 2500. Since the scattering centers in model (2) are anisotropic and polarization dependent, we apply (7) to form the image for each narrow-angle subaperture and polarization, and combine the images using equation (4). Recent approaches for joint reconstruction of multiple images [54] may also be applied to simultaneously reconstruct all polarizations for each subaperture. V. W IDE -A NGLE T OMOGRAPHIC SAR I MAGING The second approach we consider for 3D reconstruction is a tomographic SAR approach [11]–[20], in which the relative phase information from several closely spaced collection paths is used to estimate the height scattering profile using interferometric techniques. Applying this approach in combination with angle subapertures, one can divide the 3D problem problem into a set of 2D subaperture image formation problems followed by 1D spectral estimation computations. This approach results in significantly lower computation and memory requirements as compared with the method presented in Section IV. On the other hand, the Tomo-SAR-based approach applies only to multi-baseline images, and thus applies only to a particular subclass of sparse data collection geometries. As a result, the algorithm proposed in this section does not have the generality of the ℓp regularized LS approach, but does provide reduced computation for those cases in which the data collection geometry is amenable to this approach. Tomographic SAR approaches have been considered for forest canopy and building height estimation using relatively narrowangle linear collection geometries [13]–[16], [18]–[20]. Here, we adapt this approach to full-360◦ spotlight SAR data collections and demonstrate 3D vehicle reconstructions. A. Circular SAR as a Sparse Collection Aperture The CSAR system collects coherent backscatter measurements r(f ; φi , θℓ , pol) on circular apertures parameterized with azimuth angles {φi } covering [0, 2π] and at a small set of L (e.g. 2-15) elevation angles {θℓ }. The backscatter measurements, r(fj ; φi , θℓ , pol), are collected at discrete set of frequency samples {fj }. The radar measurements {r(fj ; φi , θℓ , pol)} correspond to the samples of G(kx , ky , kz ; φi , θℓ , pol) on a two-dimensional conical manifold at points kxj,n , kyj,n , and kzj,n from (3) (see Figure 2(b)), where the aspect indices (i, ℓ) map to the single index n in (4). The reconstruction problem is to estimate the threedimensional reflectivity function of the spotlighted scene



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g(x, y, z) from the set of radar returns {r(fj ; φn , θn , pol)} collected by the radar. As before, we adopt a subaperture imaging approach, and for each elevation we divide the azimuth measurements into M subapertures, each centered at azimuth φm and with fixed, user-selected azimuth extent (typically 5–20◦ for X-band data). The m-th set of subaperture data, for 1 ≤ m ≤ M , is thus a set of L elevation passes at elevation θℓ , centered at azimuth φm . Rather than store the k-space data directly, we can provide compact image products matched to scatterers with limited persistence, and maintain 1-1 correspondence with the original k-space data. These image products are 2D ground plane (z = 0) image sequences {I(x, y, 0; φm , θℓ , pol)}m where each image is the output of a filter matched to a limitedpersistence reflector over the azimuth angles in azimuth window Wm (φ). Specifically, the m-th subaperture images are constructed as I(x, y, 0; φm , θℓ , pol) = " q −1 F(x,y) G(kx , ky , kx2 + ky2 tan(θℓ ); φm , θℓ , pol)

Wm tan

−1

kx ky

#

,

−1 is the 2D inverse Fourier transform, and the where F(x,y) azimuthal window function W(φm ) is defined as: ( m , −∆/2 < φ < ∆/2 W φ−φ ∆ (8) Wm (φ) = 0, otherwise.

Here, φm is the center azimuth angle for the m-th window and ∆ describes the hypothesized azimuth persistence width. The window function W (·) is an invertible tapered window used for cross-range sidelobe reduction; typical choices may be the Hamming or Taylor windows that are commonly used in SAR images. Each image can be modulated to baseband and sampled at a lower resolution in (x, y) without causing aliasing. Each baseband ground image I B (x, y, 0; φm , θℓ , pol) is calculated as: 0

0

I(x, y, 0; φm , θℓ , pol)e−j(kx,m x+ky,m y) .

(9)

0 0 where the center frequency (kx,m , ky,m ) is determined by the center aperture φm , mean elevation angle θ¯ and center frequency fc :

4πfc 4πfc 0 cos θ¯ cos φm , ky,m = cos θ¯ sin φm . c c An important property of this subaperture imaging approach is that Nyquist sampling of (x, y) in subaperture images is dictated by the baseband downrange and crossrange k-space extents, and therefore, the image sample spacing is (much) less fine than if the full SAR image is formed using all kspace data jointly [21]. For modest azimuth window extent ∆ in radians, the Nyquist sampling in the downrange is dictated 1 , and the crossrange by the inverse of the radar bandwidth, BW 1 sampling is dictated by ∆(fc +BW/2) ; these sample spacings 0 kx,m =

B. Tomographic SAR We next present a method for using the set of ground plane images I B (x, y, 0; φm , θℓ , pol) to construct three dimensional reflectivity functions for set of subapertures centered at (φm , θℓ ). The input to the wide-angle Tomo-SAR algorithm is a set of baseband modulated ground plane images {I B (x, y, 0; φm , θℓ , pol)} at a given subaperture centered at φm of data collected at elevation cuts θℓ . We process each subaperture separately; for each subaperture denote the image sequence as {I(x, y; θℓ , pol)}L ℓ=1 and consider without loss of generality φm = 0. We consider a finite (and small) number, p, of scattering centers at each resolution cell (x, y) and reparameterize the scene reflectivity g(x, y, z) as gp (x, y) ≡ g(x, y, hp (x, y)),

1 ≤ ℓ ≤ L,

I B (x, y, 0;φm , θℓ , pol) =

1 spacing that would be are much coarser than the 2(fc +BW/2) needed for the full Circular SAR k-space data. The result is a significantly smaller storage requirement for CSAR imagery data.

(10)

where gp (x, y) denotes the complex-valued reflectivity of the scattering center at location (x, y, hp (x, y)). In general, the number of scattering centers per resolution cell varies spatially and needs to be estimated from the data. The ground plane image for elevation θℓ can can be written as I(xℓ , yℓ ; θℓ , pol) = X 0 0 s(x, y) ⋆ gp (x, y)e−j tan(θℓ )kx hp (x,y) e−jxkx ,

(11)

p

where s(x, y) is the inverse Fourier transform of the 2D windowing function used in imaging — the 2D point spread c ¯ function of the imaging operator — and kx0 = ( 4πf c ) cos(θ) is the center frequency used in baseband modulation. The ground locations (x, y, hp (x, y)) and the image coordinates (xl , yl )) are related through layover: xl = x + tan(θℓ )hp (x, y).

yl = y.

(12)

We assume that the difference between the elevation angles for the different passes is sufficiently small so that for each elevation pass the scattering center (x, y, h(x, y)), falls in the same resolution cell (xl , yl ); for practical object or scene heights radar point spread functions, and elevation diversity, this assumption is generally satisfied. Then the baseband images from each pass can be modeled as X 0 I(xl , yl ; θℓ , pol) = g˜p (xl , yl )e−jkx tan(θℓ )hp (xl ,yl ) , (13) p

P 0 −jxkx where g˜p (xl , yl ) ≡ s(x, y) ⋆ . This can p gp (x, y)e be expressed as a sum of complex exponential model X I(xl , yl ; θℓ , pol) = g˜p (xl , yl )e−jkp (xl ,yl ) tan(θℓ ) , (14) p

where the the frequency factor kp is given by kp (xl , yl ) =

¯ 4πfc cos(θ) hp (xl , yl ). c

(15)



8

In general, the elevation spacing of the L measurements in (14) is not equally-spaced. As an example, even though GOTCHA CSAR passes have a planned (ideal) equally-spaced separation of ∆θ = 0.18◦ in elevation. Actual flight paths differ from the planned paths, with the mean of eight elevation passes at 44.27, 44.18, 44.1, 44.01, 43.92, 43.53, 43.01, 43.06 degrees. In addition, the elevation varies as the aircraft circles the scene, as shown in Figure 3, so the elevation spacing changes as a function of subaperture index. Note that the only elevation angle dependence in (14) is in the phase term. We see from (14) that, for each pixel (xl , yl ), the problem of estimating the number P and heights hp of scattering centers from {I(x, y; θℓ , pol)}L ℓ=1 is one of estimating a set of complex exponentials from L measurements; see also [11]– [20], [55]. In Tomo-SAR, the number of scattering centers in a resolution cell must be estimated before spectral estimation methods can be applied to estimate height parameters. This is a model order selection problem, and different methods exist for model order selection [55]–[57]; a discussion of model order selection in the context of Tomo-SAR has been treated extensively in the literature (see e.g. [17], [20]). In a recent study [21] using CSAR X-band data of vehicles, the estimated model order was 1 in a large majority of cases, and when the model order was > 1, one dominant (large amplitude) scattering center was often seen. Thus, the complex exponential signal in (14) is sparse, with typically only 1 scattering center in the height dimension. This suggests that the estimation bias resulting from forcing the model order to be 1 may be small for a large fraction of pixels. Choosing the model order to be 1 presents a computational advantage, because for the single-exponential case, a maximum likelihood estimator of its frequency in white measurement noise is given by the peak of the Fourier transform of the data, and this Fourier transform is easy to compute. We thus adopt this model order approximation, and estimate, for each pixel (xl , yl ) the single dominant height location k1 (xl , yl ), as the peak of the Fourier transform of the L, I(xl , yl ; θℓ , pol) values for that pixel and calculate the height using (15) [12], [13], [16]. The complex amplitude of the Fourier transform at the peak provides an estimate of the amplitude of the scattering center. As previously mentioned, in this Tomographic SAR approach, the 3D reconstruction problem has been realized by 2D followed by 1D processing steps. First, 2D images are formed for each azimuth subaperture and each elevation angle. Then, 1D processing is applied to each L × 1 vector obtained by stacking the set of L elevation images and selecting the L values at a pixel location of interest. The processing reduction is afforded by the particular structure of the sparse measurement geometry provided by CSAR collections. VI. 3D I MAGING R ESULTS We next present 3D SAR image reconstruction results from both the squiggle path and the CSAR datasets, using the algorithms described in the previous two sections. We show both raw voxel reconstructed images, and smoothed surface fit reconstructions that are useful for visualization.

A. Squiggle Path Reconstructions The k-space data from the path shown in Figure 1 are first partitioned into overlapping subapertures, each with azimuth angle extent of 10◦ and full elevation extent, and separated by 5◦ center azimuth increments. As an example, Figure 4 shows the magnitude of k-space data from the k-space subset in azimuth range of [66◦ , 76◦ ).

Fig. 4. Magnitude of k-space data subset from azimuth range [66◦ , 76◦ ). Lighter colors and smaller points are used for smaller magnitude samples; darker colors and larger points are used for larger magnitude samples. Axes units are in rad/m.

Each subset of data is contained in a bounding box with bandwidths in each dimension of (XBW , YBW , ZBW ) = (142.80, 314.2, 285.6) rad/m. At these bandwidths, spatial samples are critically sampled with sample spacings of (∆x, ∆y, ∆z) = (0.044, 0.02, 0.022) meters in each respective dimension. Both the image reconstruction and k-space interpolation are performed on uniformly spaced 182×250×252 grids. With this size grid, the spatial extent of the reconstructed images is [−4, 4) × [−2.5, 2.5) × [−2.77, 2.77) meters in the x, y, and z dimensions respectively. Each subset of k-space data is interpolated using nearest neighbor interpolation. In simulations not presented here, more accurate interpolations using both the Epanechnikov and Gaussian kernels, were found to result in nearly identical images, but at much higher computational cost. The squiggle path dataset is noiseless. To simulate the effect of radar measurement noise, we corrupt the k-space data with i.i.d. circular complex Gaussian noise with zero mean and variance, σn2 = 0.9. Real and imaginary parts of the k-space data have a mean of approximately zero and variance, σs2 , of approximately 9; thus, the noise variance is chosen so that the signal to noise ratio (SNR) is 10 dB, where SNR in decibels σ2 is defined as 10 log( σ2s ). n First, we show in Figure 5 a side view of a ’gold standard’ benchmark 3D reconstructed backhoe image corresponding to the squiggle path dataset [45]. The image was formed using a windowed 3D inverse Fourier transform of a dense kspace dataset covering the azimuth and elevation range of the 1 ◦ squiggle path; this dense data is given for every 14 in azimuth and elevation angle along an azimuth range of [65.5◦ , 114.5◦ ] and elevation range of [17.5◦ , 42.5◦ ]. Squiggle path k-space data is contained within this benchmark dataset and is very sparse with respect to it; see Fig. 1. The squiggle path dataset consists of only 1.29% of the benchmark data samples.



9

(a)

(b) Fig. 5. Benchmark reconstructed backhoe image using k-space data collected 1 ◦ at every 14 in azimuth and elevation angle along an azimuth range of [65.5◦ , 114.5◦ ] and elevation range of [17.5◦ , 42.5◦ ]. Subfigure (a) displays the reconstructed image superimposed on the backhoe facet model, and (b) shows the reconstructed image without the facet model. Images from [45].

Second, we present image reconstructions from the squiggle path using standard Fourier image reconstruction. A reconstructed squiggle path image viewed from the side and top is shown in Figure 7. The top 25 dB magnitude voxels are displayed in the image. One sees that the structure of the backhoe is highly smoothed and distorted due to the high image sidelobes, and backhoe features, such as the front scoop are not well localized. Poor image quality is predicted from the subaperture point spread functions of the sparse squiggle path; an example is shown in Figure 6 for one subaperture. The PSF is not well localized and exhibits significant spreading and high sidelobes due to the sparseness of the data.

Fig. 6. Magnitude of PSF from the squiggle path over azimuth range [66◦ , 76◦ ). Light colors and small points are used for small magnitude voxels; darker colors and large points are used for large magnitude voxels. Axis units are in meters.

Figure 8 shows the side and top view of a reconstructed squiggle path backhoe image using the ℓp regularized LS reconstruction algorithm in Section IV. The top 30 dB magnitude voxels are displayed. The images in Figure 8 were formed by first reconstructing 27, 3D images from each subaperture and polarization; images are the solution to the optimization problem (7). All images are reconstructed using a norm with p = 1 and sparsity parameter λ = 10, which are selected manually. Automatic selection of λ is an ongoing area of research [58]–[60]. Here, p, and λ were chosen empirically through visual inspection of images. Final images are formed by combining the subset images over the maximum of polarizations in addition to aspect angles according to (4). In addition to the scattering point plots displayed in the top of Figure 8, it is possible to accentuate surfaces of 3D reconstructed images for visualization by smoothing image voxels; visualizations are shown in Figures 8(e) and 8(f) 1 . There are a large array of scientific visualization tools for accomplishing such a task, such as Maya and ParaView. Maya visualization examples are given in [61]. Here we apply a Gaussian kernel with diagonal covariance and equal standard deviation, σ, to smooth the voxels. Smoothed images are formed on a grid with the same dimensions as the original grid. To speed up the smoothing, the kernel is given a fixed support within some radius of the grid voxel being smoothed. In Figures 8(e) and 8(f), a standard deviation of σ = 0.4 m and grid radius of 3σ is used. Voxel magnitude is then displayed using color and transparency coding. Blue, transparent colors indicate low relative voxel magnitude and red, opaque colors indicate large relative voxel magnitude. As can be seen from Figure 8, features in the sparse reconstructions are well-resolved. For example, the hood, roof, and front and back scoops are clearly visible, in the correct location, and do not exhibit the large sidelobe spreading seen in Figure 7. The side panels of the driver cab are not visible, and the arm on the back scoop is not as prominent as in the benchmark in Figure 5, but most backhoe features in the benchmark backhoe image are also visible in the squiggle path reconstruction. There are a small number of artifacts in the image that do not lie close to the backhoe, namely below the front and back scoop. These artifacts appear to be due to multiple-bounce effects that are present in the given scattering data, rather than to an ‘error’ artifact of the reconstruction process. From the top view of the backhoe, the group of voxels at the top left also appear to be present in the benchmark image as viewed from an angle not shown in Figure 5; these voxels are also likely the result of multibounce from the back scoop and are not artifacts specific to squiggle path reconstruction. Simulation results presented above were performed in MATLAB on a system with an Intel 3 GHz Dual Core Xeon processor and 4 GB of memory. Both the interpolation and sparse optimization in image reconstruction can be computationally intensive. The Nearest-neighbor interpolation method 1 A movie of this visualization rotating 360◦ is included in the multimedia file bh_squiggle_vis.avi, which is available for download on http://ieeexplore.ieee.org . During rotation of the backhoe, one side appears more filled than the other due to limited azimuth data collection extent.



10

(a)

(b)

(c)

(d)

Fig. 7. Reconstructed backhoe from standard Fourier image reconstruction using each subaperture image for an SNR of 10 dB. Lighter colors and smaller points are used for smaller magnitude voxels; darker colors and larger points are used for larger magnitude voxels. Subfigure (a) and (b) show a side view of the reconstructed image with and without the backhoe facet model superimposed, respectively; subfigures (c) and (d) show top views of the reconstructed image with and without the backhoe facet model superimposed, respectively. The top 25 dB magnitude image voxels are displayed.

is fast compared to ℓp regularized LS optimization and took less than 25 seconds to run on each data subset; sparse optimization computations took 17−26 minutes to run on each subaperture. Although not investigated here, it may be possible to alter stopping criterion tolerances in the algorithm to lower computation times without adversely affecting reconstructed images. B. Mulitpass CSAR Reconstructions We next consider 3D vehicle reconstructions from measured X-band CSAR data taken over an urban area. Figure 9 shows a 2D radar ground image obtained from one pass of the CSAR scene. This is the scene of a parking lot with several vehicles, including a calibration tophat and a Toyota Camry. Figure 10 shows photographs of the tophat and Camry. Radar flight location information for the GOTCHA dataset contains sensor location errors. These errors are corrected using prominent-point (PP) autofocus [10] solution provided with the GOTCHA dataset; in addition, spotlighting is used to reduce computation and memory requirements; these processes are discussed in more detail in [32]. We form 3D reconstructions of two spotlighted areas of the CSAR GOTCHA scene centered on the tophat, and on the Toyota Camry. For the ℓp regularized LS reconstructions, 5◦ subapertures from 0◦ to 360◦ with no overlap were used, for a total of 72 subaperture images that are combined by (4). Reconstructed ℓp regularized LS image voxels are spaced at 0.1 m in all three dimensions. The dimensions of the reconstructed tophat and Camry images in (x, y, z) dimensions are [−2, 2) × [−2, 2) × [−2, 2) and [−5, 5) × [−5, 5) × [−5, 5) meters respectively. These

dimensions define the k-space bandwidth of the bounding box and grid used for nearest-neighbor interpolation. The bounding box bandwidth used in both images is 62.8318 rad/m in all dimensions. The interpolation grid inside the bounding box consists of 50 samples for the tophat and 100 samples for the Camry in each dimension. As before, we chose p and λ manually to generate images that produce qualitatively good reconstructions.

Fig. 9.

2D SAR image of the GOTCHA scene. Image from [46].

Figure 11 shows 3D reconstructions of the tophat and Camry formed using traditional Fourier reconstruction techniques on each interpolated subaperture dataset, and then by



11

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 8. Reconstructed backhoe image using regularized LS for an SNR of 10 dB. The top 30 dB magnitude image voxels are displayed. In (a) through (d), lighter colors and smaller points are used for smaller magnitude voxels; darker colors and larger points are used for larger magnitude voxels. In (e) and (f) smoothed visualizations are displayed. The left column of subfigures show a side view of the reconstructed image. In (a) the backhoe facet model is superimposed; The right column of subfigures show top views of the reconstructed image. In (b), the backhoe facet model is superimposed.

combining the subaperture reconstructions using (4). The VV polarization channel is used, and only the top 20 dB of voxels are shown, with lighter colors and smaller points indicating lower magnitude scattering and larger points with darker color indicating larger magnitude scattering. These images are very similar to ones generated using filtered backprojection processing. The images have poor resolution, especially in the slant plane height directions, due to the sparse support of k-space data in elevation angle; the support window of this collection geometry results in a point spread function with spreading and high sidelobes [62]. Figure 12 shows three different views of the tophat 3D reconstruction using the ℓp regularized LS approach. These

reconstructions use the VV polarization data, with parameter settings of λ = 0.01 and p = 1, and, in contrast to the Fourier images, the top 40 dB of voxels are shown. The reconstruction in Figure 12 clearly shows the circular ‘corner’ between the base and cylinder of the tophat (see Figure 10(a)), and this scattering is well-localized to the correct location. From the reconstruction, the radius of the tophat is seen to be approximately 1 m, agreeing with the true radius of the tophat. Furthermore, there are no visible artifacts in the image. Figure 13 shows ℓp regularized LS reconstructions of the Toyota Camry for two polarizations (VV and HH). The parameters λ = 10, and p = 1 are used in the reconstructions, and the top 40 dB of scattering centers are shown. To highlight



12

(a) Tophat

(b) Camry Fig. 10.

Photographs from the GOTCHA scene. Images from [46].

virtual dihedral made up of the ground and the vertical vehicle sides, front, and back. The HH images appear to show more scattering from this virtual dihedral than to the VV images, as there is a more pronounced line below the car; there is also some scattering above the windshield in the HH image, which may be an artifact, and does not appear in the VV image. The apparent artifacts in the VV polarization below the front of the car and to the side of the car in the 3D view, are scattering from an adjacent vehicle that is not completely removed by the spotlighting process. In Figure 14, we illustrate the aspect dependence of the proposed ℓp regularized LS non-coherent imaging process. Whereas, previous figures were color-coded on voxel magnitude, Figure 14 is color-coded on azimuth angle. The color of a voxel indicates center azimuth angle of the subaperture image that it came from. The circle at the base of the Camry shows azimuth angle of the aircraft with respect the the Camry. Computations for Figures 12-14 were performed in MATLAB on a system with an Intel 2.8 GHz Pentium D processor and 2 GB of memory. Interpolation time with nearest-neighbor interpolation was negligible; sparse optimization computations took 3 − 5 minutes on each subaperture.

(a) Tophat

(a) 3D view

(b) Camry Fig. 11.

Traditional Fourier images (b) Side view

vehicle structure, images are displayed using the smoothing visualization process as described in Section VI-A with a Gaussian kernel standard deviation of σ = 0.1 m. An example of a non-smoothed scatter point plot of the Camry is given in Figure 14. Figures 13(g) through 13(i) 2 show combined HH and VV polarization images formed by taking the maximum over polarizations in (4) in addition to aspect angle. In all of the images, the outline of the Camry is clearly visible. The upper, curved line is direct scattering from the vehicle itself, whereas the lower curve at 0 m elevation is scattering from the 2A

movie of the combined VV and HH polarization visualization rotating 360◦ is included in the multimedia file sparse_Camry_HH_VV_vis.avi, which is available for download on http://ieeexplore.ieee.org.

(c) Top view Fig. 12. ℓp regularized LS tophat reconstructions with λ = 0.01 and p = 1. The top 40 dB magnitude voxels are shown.



13

Fig. 13.

(a) 3D view, VV polarization

(b) Side view, VV polarization

(c) Top view, VV polarization

(d) 3D view, HH polarization

(e) Side view, HH polarization

(f) Top view, HH polarization

(g) 3D view, VV and HH polarization

(h) Side view, VV and HH polarization

(i) Top view, VV and HH polarization

ℓp regularized LS Camry reconstructions with λ = 10 and p = 1. The top 40 dB magnitude voxels are shown.

In the Tomo-SAR approach, reconstructed scattering centers are not constrained to lay on a grid in the height dimension. To compare this imaging method with ℓp regularized LS reconstructed images, data is first interpolated to a grid with 0.1 m voxel spacing in each dimension; this is the same spacing used in ℓp regularized LS reconstructions. A Gaussian kernel with standard deviation of σ = 0.1 m is used for interpolation. Figure 15 shows the results of the Tomo-SAR approach applied to the Camry data after interpolation. The top 20 dB points are shown. The VV and HH polarization images in Figure 15(g) through 15(i) 3 are formed by combining the interpolated VV and HH polarization images as performed in ℓp regularized LS reconstructions. Scattering is assumed to be above the ground plane in calculations; so, unlike in the ℓp regularized LS reconstruction, there are no non-zero voxels below the vehicle. As in the ℓp regularized LS reconstruction, a set of 72 subaperture image sets were formed, each 3A movie of the combined VV and HH polarization visualization rotating 360◦ is included in the multimedia file sparse_Camry_HH_VV_vis_Tomo_SAR.avi, which is available for download on http://ieeexplore.ieee.org.

with 5◦ azimuth extent, and the image-domain subaperture reconstructions for all polarizations were combined using (4). The wide-angle Tomo-SAR algorithm was also implemented in MATLAB and took less than 1 minute to process each subaperture. In comparing the ℓp regularized LS and Tomographic SAR reconstructions, some qualitative differences are seen. Most notably, the Tomo-SAR-based reconstructions are more filled than the regularized LS reconstructions. This is in large part due to the way in which sparsity is enforced in the two techniques; the ℓp regularized LS method imposes sparsity in the full 3D space, while Tomo-SAR-based methods obtain standard (non-sparse) 2D images and develop sparse reconstructions only in the 1D height dimension. The 2D image downrange and crossrange resolutions are approximately 0.3 meters and 0.2 meters, respectively; so, a single bright scattering point will appear as a 0.3m × 0.2m flat disk, tilted at 45◦ . For 3D visualizations, we find the more filled Tomo-SAR reconstructions to be more easily interpretable. For automated post-processing such as automatic target recognition that treats the reconstructed voxels as features, the smaller number of ’features’ provided by the ℓp regularized LS approach is likely



14

Fig. 15.

(a) 3D view, VV polarization

(b) Side view, VV polarization

(c) Top view, VV polarization

(d) 3D view, HH polarization

(e) Side view, HH polarization

(f) Top view, HH polarization

(g) 3D view, VV and HH polarization

(h) Side view, VV and HH polarization

(i) Top view, VV and HH polarization

3D Camry reconstruction using the wide-angle Tomo-SAR algorithm. The top 20 dB magnitude scatterers are shown

preferable, since it results in less correlated features than in the Tomo-SAR technique. In comparing computations, we see that the ℓp approach requires more computation time for 3D reconstructions than the Tomo-SAR approach does, in the present algorithmic implementations. It should be noted that we have not undertaken a dedicated effort at computation optimization, and different relative computation times may be achieved with additional optimization. Finally, we note that the ℓp regularized LS 3D SAR images do indeed show significant sparsity, which provides further justification of the validity of enforcing a model order of 1 in the Tomo-SAR reconstructions, as discussed in Section V. VII. C ONCLUSIONS

reconstruction. Two reconstruction approaches were presented. The first uses a sparsity-constrained regularized least-squares technique to directly compute 3D reconstructions from arbitrary collection geometries. The algorithm is demanding in both computation and memory usage. A second Tomo-SAR approach is tailored to a particular sparse data collection: a multi-elevation Circular SAR collection geometry. This latter approach takes advantage of the particular data collection geometry to partition a 3D reconstruction problem into a set of 2D image formation steps followed by 1D height estimates, yielding savings in both memory and computation. Both methods are effective at significantly reducing the large sidelobe artifacts that are present in traditional Fourier-based or backprojection reconstruction methods.

We have examined the use of scattering sparsity to improve 3D SAR reconstruction from sparse data collection geometries. We have formulated 3D reconstruction algorithms based on the premise that radar scattering is sparse in the reconstructed 3D spatial domain. The algorithms consider anisotropic scattering behavior of objects over wide aspect angles, but uses a GRLT-based approach to noncoherently combine independently calculated subaperture images to obtain a wide-angle

We presented 3D image reconstructions using both synthetic backscatter measurements of a construction backhoe and Circular SAR X-band radar measurements of an urban ground scene. In the backhoe case, we presented 3D reconstructions using a pseudorandom “squiggle” flight path that is sparse over a wide-angle aperture in both azimuth and elevation; the sparse flight path includes only 1.29% of the filled-aperture data in the same azimuth-elevation sector. The resulting re-



15

component index of the x vector, and ∗ θ(xi , xni ) =|xni |p + Re p (xni ) |xni |p−2 (xi − xni ) 1 + p|xni |p−2 |xi − xni |2 . 2 It was shown in [28] that the sequence of solutions

¯ xn ) argmin J(x, x −1 λ = AH A + D(xn ) AH y, (18) 2 where D(xn ) = diag p|xni |p−2 , converges to a solution to (7) as n → ∞. For the imaging problems considered here, direct inversion of the matrix in (18) can be computationally intensive, and we utilize the the conjugate gradient (CG) method to solve the inverse. The algorithm decomposes into a nested loop. The outer loop iterates on the solution xn , and the inner loop is the CG loop that solves the inverse in (18). To arrive at an exact solution to the original optimization problem, the outer loop must be executed an infinite amount of times. Here, we terminate the outer loop when the relative change in the original objective function is small between iterations, and we terminate the inner CG loop when the relative magnitude of the residual becomes small. The tolerances used here for algorithm termination were qualitatively chosen. These tolerances affect image quality, and execution speed, but empirically there does not appear to be much improvement in image quality by decreasing tolerance past a certain level. xn+1

(a) 3D view, VV and HH polarization

(b) Top view, VV and HH polarization Fig. 14. ℓp regularized LS Camry azimuth angle color-coded reconstructions using combined VV and HH polarizations with λ = 10 and p = 1. The top 40 dB magnitude voxels are shown. Colobar units are in degrees.

(17)

=

R EFERENCES construction exhibits better resolution and far lower sidelobes than a conventionally-formed reconstruction, and it compares favorably with a reconstruction obtained using filled aperture data in the same azimuth-elevation sector. In addition, we presented reconstruction results for two ground objects (a calibration tophat and a Toyota Camry) from measured Circular SAR data. Both the regularized least-squares and the TomoSAR reconstruction techniques were applied. Both algorithms were able to produce 3D reconstructions that clearly show the shape of the ground objects, with significantly lower sidelobe artifacts than those obtained in Fourier or backprojection imagery. A PPENDIX In this appendix, we review the monotonic iterative algorithm presented in [28] used for to solve the ℓp regularized LS optimization problem (7). The algorithm utilizes an optimization transfer technique, where sequence of functions that majorize (7) is optimized, and the sequence of optimum solutions converges to the solution of the original optimization problem. Define the sequence of cost functions ¯ xn ) = ky − Axk2 + λ J(x, 2

N X

θ(xi , xni ),

(16)

i=1

where superscript n is the sequence index; subscript i is the

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[34] K. Naidu and L. Lin, “Data dome: full k-space sampling data for highfrequency radar research,” in Algorithms for Synthetic Aperture Radar Imagery XI. Orlando, FL.: SPIE Defense and Security Symposium, April 12–16 2004. [35] C. H. Casteel, L. A. Gorham, M. J. Minardi, S. Scarborough, and K. D. Naidu, “A challenge problem for 2D/3D imaging of targets from a volumetric data set in an urban environment,” in Algorithms for Synthetic Aperture Radar Imagery XIV, E. G. Zelnio and F. D. Garber, Eds. Orlando, FL.: SPIE Defense and Security Symposium, April 9–13 2007. [36] D. E. Dudgeon, R. T. Lacoss, C. H. Lazott, and J. G. Verly, “Use of persistant scatterers for model-based recognition,” in Algorithms for Synthetic Aperture Radar Imagery (Proc. SPIE 2230), D. A. Giglio, Ed., 1994, pp. 356–368. [37] R. Bhalla, J. Moore, and H. Ling, “A global scattering center representation of complex targets using the shooting and bouncing ray technique,” IEEE Trans. on Antennas and Propagation, vol. 45, no. 6, pp. 1850– 1856, 1997. [38] D. Rossi and A. Willsky, “Reconstruction from projections based on detection and estimation of objects–Parts I and II: Performance analysis and robustness analysis,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 32, pp. 886–906, 1984. [39] R. L. Moses, E. Ertin, and C. Austin, “Synthetic aperture radar visualization,” in Proceedings of the 38th Asilomar Conference on Signals,Systems, and Computers, Pacific Grove, CA, Nov 2004. [40] K. E. Dungan and L. C. Potter, “Classifying sets of attributed scattering centers using a hash coded database,” in Algorithms for Synthetic Aperture Radar Imagery XVII, E. G. Zelnio and F. D. Garber, Eds. Orlando, FL.: SPIE Defense and Security Symposium, April 5–9 2010. [41] ——, “Classifying transformation-variant attributed point patterns,” Pattern Recognition, vol. 43, no. 11, pp. 3805–3816, November 2010. [42] K. Varshney, M. Çetin, J. Fisher, and A. Willsky, “Sparse representation in structured dictionaries with application to synthetic aperture radar,” IEEE Transactions on Signal Processing, vol. 56, no. 8, pp. 3548–3561, August 2008. [43] I. Stojanovic, M. Çetin, and W. C. Karl, “Joint space-aspect reconstruction of wide-angle SAR exploiting sparsity,” in Algorithms for Synthetic Aperture Radar Imagery XV. Orlando, FL.: SPIE Defense and Security Symposium, March 17–18 2008. [44] J. A. Jackson and R. L. Moses, “An algorithm for 3D target scatterer feature estimation from sparse SAR apertures,” in Algorithms for Synthetic Aperture Radar Imagery XVI (Proc. SPIE vol. 7337), E. G. Zelnio and F. D. Garber, Eds., 2009. [45] Air Force Research Laboratory. (2010, January) Backhoe sample public release and Visual-D challenge problem. [Online]. Available: https://www.sdms.afrl.af.mil/request/data request.php#Visual-D [46] ——. (2010, January) Gotcha 2D / 3D imaging challenge problem. [Online]. Available: https://www.sdms.afrl.af.mil/datasets/gotcha/ [47] E. Candés and J. Romberg, “ℓ1 -MAGIC: Recovery of sparse signals via convex programming,,” California Institute of Technology, Tech. Rep., October 2005. [48] A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sciences, vol. 2, no. 1, pp. 183–202, January 2009. [49] I. Daubechies, M. Defrise, and C. D. Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Comm. Pure Appl. Math., vol. 57, no. 11, pp. 1413–1457, 2004. [50] T. Goldstein and S. Osher, “The split Bregman method for L1regularized problems,” SIAM Journal on Imaging Sciences, vol. 2, no. 2, pp. 323–343, 2009. ¨ ur Yilmaz, “Stable sparse approxima[51] R. Saab, R. Chartrand, and Ozg¨ tions via nonconvex optimization,” in 33rd International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2008. [52] L. Greengard and J.-Y. Lee, “Accelerating the nonuniform Fast Fourier Transform,” SIAM Review, vol. 43, no. 3, pp. 443–454, 2004. [53] J. Fessler and B. Sutton, “Nonuniform Fast Fourier Transforms using min-max interpolation,” IEEE Transactions on Signal Processing, vol. 51, no. 2, pp. 560–574, February 2003. [54] N. Ramakrishnan, E. Ertin, and R. Moses, “Enhancement of coupled multichannel images using sparsity constraints,” IEEE Transactions on Image Processing, vol. 19, no. 8, pp. 2115–2126, August 2010. [55] P. Stoica and R. Moses, Spectral Estimation of Signals. Prentice Hall, 2005. [56] M. Wax and T. Kailath, “Detection of signals by information theoretic criteria,” IEEE Trans. ASSP, vol. 33, pp. 387–392, April 1985. [57] D. N. Lawley, “Tests of significance of the latent roots of the covariance and correlation matrices,” Biometrica, vol. 43, pp. 128–136, 1956.



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[58] D. Malioutov, M. Çetin, and A. Willsky, “Homotopy continuation for sparse signal representation,” in Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP 05), vol. 5, March 2005, pp. 733–736. ¨ Batu and M. C [59] O. ¸ etin, “Hyper-parameter selection in non-quadratic regularization-based radar image formation,” in Algorithms for Synthetic Aperture Radar Imagery XV. Orlando, FL.: SPIE Defense and Security Symposium, March 17–20 2008. [60] C. Austin, R. Moses, J. Ash, and E. Ertin, “On the relation between sparse reconstruction and parameter estimation with model order selection,” IEEE Journal of Selected Topics in Signal Processing, vol. 4, no. 3, pp. 560 –570, June 2010. [61] R. Moses, P. Adams, and T. Biddlecome, “Three-dimensional target visualization from wide-angle IFSAR data,” in Algorithms for Synthetic Aperture Radar Imagery XII. Orlando, FL.: SPIE Defense and Security Symposium, March 28 – April 1 2005. [62] E. Ertin, C. D. Austin, S. Sharma, R. L. Moses, and L. C. Potter, “GOTCHA experience report: Three-dimensional SAR imaging with complete circular apertures,” in Algorithms for Synthetic Aperture Radar Imagery XIV, E. G. Zelnio and F. D. Garber, Eds. Orlando, FL.: SPIE Defense and Security Symposium, April 9–13 2007.

Randolph L. Moses (S’78-M’85-SM’90) received the B.S., M.S., and Ph.D. degrees in electrical engineering from Virginia Polytechnic Institute and State University in 1979, 1980, and 1984, respectively. During summer 1983, he was a SCEEE Summer Faculty Research Fellow with Rome Air Development Center, Rome, NY. From 1984 to 1985, he was with the Eindhoven University of Technology, Eindhoven, The Netherlands, as a NATO Postdoctoral Fellow. Since 1985, he has been with the Department of Electrical Engineering, The Ohio State University, Columbus, and is currently a professor there, and serves as Director for the Institute for Sensing Systems. From 1994 to 1995, he was on sabbatical leave as a visiting researcher with the System and Control Group, Uppsala University, Sweden. His research interests are in time series analysis, radar signal processing, sensor array processing, and sensor networks. Dr. Moses is an Associate Editor for the IEEE TRANSACTIONS ON IMAGE PROCESSING and serves on the Sensor Array and Multichannel (SAM) Technical Committee of the IEEE Signal Processing Society. He is a member of Eta Kappa Nu, Tau Beta Pi, Phi Kappa Phi, and Sigma Xi.

Christian D. Austin (S’02) received a B.E. degree in Computer Engineering and a B.S. degree in Mathematics from the State University of New York (SUNY) at Stony Brook in 2003. In 2006, he received his M.S. in Electrical Engineering from The Ohio State University, Columbus, Ohio. Currently, he is pursuing a Ph.D. degree in Electrical Engineering at the Ohio State University. His research interests include statistical signal processing, compressive sensing, and synthetic aperture radar.

Emre Ertin is a Research Assistant Professor with the Department of Electrical and Computer Engineering at the Ohio State University. He received the B.S. degree in Electrical Engineering and Physics from Bogazici University, Turkey in 1992, the M.Sc. degree in Telecommunication and Signal Processing from Imperial College, U.K. in 1993, and the Ph.D. degree in Electrical Engineering from the Ohio State University in 1999. From 1999 to 2002 he was with the Core Technology Group at Battelle Memorial Institute. His current research interests are statistical signal processing, wireless sensor networks, radar signal processing, biomedical sensors, distributed optimization and control.
