Embedded Second-Order Cone Programming with Radar Applications

Embedded Second-Order Cone Programming with Radar Applications Paul Mountcastle∗ , Tom Henretty† , Aale Naqvi‡ and Richard Lethin§

Reservoir Labs, Inc. New York, NY 10012 Email: ∗ [email protected], † [email protected], ‡ [email protected], § [email protected] Abstract—Second-order cone programming (SOCP) is required for the solution of underdetermined systems of linear equations with complex coefficients, subject to the minimization of a convex objective function. This type of computational problem appears in compressed radar sensing, where the goal is to reconstruct a sparse image in a generalized space of phase model parameters whose dimension is higher than the number of complex measurements. In order to enforce sparsity in the final rectified radar image, the sum of moduli of a complex vector, called the 1 norm, must be minimized. This norm differs from what is ordinarily encountered in compressed sensing for digital photographic data and video, in that the convex optimization that must be performed involves an SOCP rather than a linear program. We illustrate the role of this type of optimization in radar signal processing by means of examples. The examples point to a significant generalization that encompasses and unifies a wide class of radar signal processing algorithms that can be implemented in software by means of SOCP solvers. Finally, we show how modern SOCP solvers are optimized for efficient solution of these problems in the context of embedded signal processing on small autonomous platforms.

I. I NTRODUCTION Underdetermined complex linear inverse problems appear in radar imaging whenever the number of points in the image to be computed is greater than the number of measurements. This is the case in compressed sensing, where the received signal contains fewer degrees of freedom than the Nyquist criterion would require for classical Fourier reconstruction of the needed image. Compressed sensing concepts arise naturally in the context of digital photographic images. An uncompressed video frame consisting of several MP of data, each pixel comprising three color words of some depth, contains far more raw data than is really needed to caspture the full information content in the visual scene. In digital image compression (eg. JPEG compression [1]), special properties of the image itself allow it to be reconstructed perfectly from much less information. It is reasonable to ask whether this process might not provide a means of acquiring and reconstructing images exactly, while collecting only some of the pixels, or a selected set of linear combinations of pixel intensities. When the special properties This work is sponsored in part by DARPA MTO as part of the Power Efficiency Revolution for Embedded Computing Technologies (PERFECT) program (issued by DARPA/CMO under Contract No: HR0011-12-C-0123). The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressly or implied, of the DARPA or the U.S. Government. Distribution Statement A (Approved for Public Release, Distribution Unlimited).

978-1-4673-9286-0/15/$31.00 ©2015 IEEE

of the image that make it compressible can be expressed in terms of sparsity of the image vector in some linear transform domain, this is indeed the case [2]. The situation in radar imaging is that, even without any front-end downsampling, the number of measurements almost always fails the Nyquist criterion, except in the most elementary situations. This is true even when all of the available radar data is collected and processed. At the same time, sparsity is guaranteed on physical grounds as long as the dimension of the projective phase space is large compared to the dimension of the measurement. This follows from the principle of stationary phase, since the locus of stationary phase points in the image space constitutes an embedded manifold of low intrinsic dimension. Therefore radar signal processing, despite the additional technical challenge of requiring SOCP, deserves special consideration as a practical application for ideas of sparse image reconstruction. The first purpose of this paper is to present the fundamental concepts of radar signal processing in a generalized form that goes beyond the current state of the art in this field,1 emphasizing the significant role that SOCP solvers play in realizing the promise of this technology. The second purpose is to review some of the optimizations that we employ to make these particular computations amenable to deployment on small, low-power, embedded platforms. The rest of the paper is organized as follows. In Section II, we provide additional technical background on the role of SOCP in compressive radar signal processing, and also discuss the current state of the art in embedded SOCP solvers. In section III, we provide a review of several compressive radar applications. These applications serve to motivate our presentation of a broad and versatile formulation of the radar sensing problem and its solution that we call Generalized Compressive Processing (GCP). GCP uses SOCP to perform imaging in an abstract space that consists of model parameters called generalized coordinates that fix the two-way radar phase. In Section IV, we discuss the software techniques used by Reservoir Labs to accelerate the solution of state-of-theart SOCP solvers by orders of magnitude, bringing the signal processing requirements for our algorithms into line with the size, weight and power requirements of small embedded radar platforms like UAVs. We conclude with a brief review of 1 Patents

pending.

related work and a foreword to future work in this area in Section V. II. BACKGROUND A. Generalized Compressive Processing Modern radar signal processing focuses on the phase change of a sinusoidal signal as it propagates from the radar to the target scatterers and back again. The radar itself may consist of some elements in a certain geometrical configuration counted by a discrete index e ∈ {1 . . . NE }, a set of Nyquist frequencies (if the signal has bandwidth) counted by a discrete index f ∈ {1 . . . NF }, and a set of pulse times (for a typical transmitted signal that consists of short pulses) counted by a discrete index t ∈ {1 . . . NT }. A few examples that illustrate this idea will be given in Section III. The three indices being discrete and finite, it is always possible to arrange them into a single long vector n ∈ {1 . . . NE · NF · NT } by putting the tuples (e, f, t) in some order. For stating the general principle of compressed radar processing, this order is unimportant. Of course it must be remembered when setting up a particular calculation. We will not be concerned with nonlinear media for now, so each frequency of the signal can be treated as though it propagated independently. Then the information about targets is contained in the linear transformation that is experienced by each of the frequencies. It is characterized by an amplitude attenuation and a phase shift that can be written in polar form and arranged into a complex vector z (n) = z (f, e, t). The unique attenuations and phase shifts of the signal at each element, wideband frequency and pulse in the signal constitutes all of the measured information about scatterers in the environment. We introduce a model φ(f, e, t|q) = φ(n|q) of the phase shift corresponding the scatterers we would like to selectively image. The model is determined in each particular case by the physics. This model, besides referring to the discrete indices (f, e, t) contains some additional model parameters q. These model parameters can be considered to be the coordinates of an abstract multidimensional image space called path space, with the entire set of required parameters indicated by a single letter q. This set of parameters are called generalized coordinates to highlight their close relation to mechanical coordinates of motion. We call the space of generalized coordinates path space, because each point in this space corresponds to one path of a hypothetical target scatterer consistent with the model parameters q. We will introduce a complex n-dimensional column vector with elements n|z = z (n) and a corresponding row vector called the steering vector with elements q|n = ei φ(n|q) . The term steering vector generalizes an idea from array beamsteering that will be given as one example in Section III-A. In the general context, the steering vector picks out one point in the path space q that we want to illuminate. We can now introduce a real, positive field σ(q) over the path space

978-1-4673-9286-0/15/$31.00 ©2015 IEEE

q as follows

 2    −φ(n|q)  σ (q) = q | z z | q =  z (n) e   

(1)

n

This quantity, when normalized over all of the points in the space q, can be interpreted as the likelihood that a scatterer with generalized coordinates q is present in the scene. It is also possible to introduce real or complex weights in the definition of σ(q), in order to optimize additional image criteria. In the interest of brevity, these ideas of probability and adaptive weights will not be covered in this paper. They are treated in detail in Reference [3]. To see the role of convex optimization in compressive radar computations, we first imagine the path space of generalized coordinates to be populated with a set of discrete sampling points qk , where k ∈ {1 . . . K} is an index running over the sample points. The concept of a discrete sampling in the path space q applies to any radar signal processing problem where it is possible to write the phase as a function of the measurement indices and a set of generalized coordinates. If n ∈ {1 . . . N } is the single index that runs over all of the measurement dimensions, we can identify a complex N × K matrix A with elements Ank = e−iφ(n|qk ) .

(2)

Define a complex column vector Φ of length K as 2

σ (qk ) = |Φ (qk )| = Φ∗k Φk .

(3)

zk (n) = Φk e+iφ(n|qk )

(4)

With these definitions, Equation 1 can be rewritten Az = Φ. In this form, we see that A is analogous to the discrete Fourier transform matrix. The significant generalizations are (1) The phase is no longer required to be affine in the generalized coordinates q, and (2) The number of sample points K is not related to the number of measurements N . In a typical case, there are more generalized coordinates than measurement dimensions, so the mapping from measurements to image is not a transform in the ordinary sense. It is a one-tomany mapping. Nonetheless, it has been possible to recast the imaging equation in the form of a matrix-vector multiply. It is possible to adopt a completely different view of imaging that is more closely related to tomography than to Fourier analysis. For this purpose, we picture the values Φk of the complex image at a set of sample points to physically represent moving point scatterers in the scene. In other words, we replace our original concept of the radar scene as a transformation of electromagnetic signals with the concept of the scene as a set of delta functions in the path space q. The aim of imaging, in this view, is to estimate the values of the model parameters Φk . The nth sensor measurement zk (n) that would result from the presence of a delta function Φk , viewed as a bit of complex RCS at generalized image position qk , is At the same level of approximation that we have adopted all along (single scattering from each point without second

bounce) we model the measurements zn as the sum of scattering from each of the delta functions at points qk . Equating the actual measurement vector zn with this tomographic model, we have (5) z = A† Φ,

a very fast solver based on Alternating Direction Method of Multipliers (ADMM) that has built-in capability for warmstart, and is amenable to other optimizations that will be discussed in Section IV.

where the symbol A† denotes the complex conjugate transpose of the generalized DFT matrix given by Equation 2. Equation 5 has the form of a system of complex linear equations, one equation for each measurement zn . This system of equations is already under-determined in cases of practical interest, and it will be more so if some of the measurements |z are not present.

Some examples of generalized compressive processing are discussed. The purpose of these examples is to provide a more concrete mental picture of the path space, the steering vector and other concepts that have so far been introduced only as abstract definitions.

B. SOCP Solvers In many cases of interest, it is possible to obtain the sparsest solution vector Φ that is consistent with Equation 5 by minimizing the 1 -norm σ1 of the solution subject to the linear equations, taken as constraints. Alternately, a Lagrangian  linear combination of the root-sum-square error z − A† Φ2 and σ1 can be minimized, the Lagrange multiplier playing the role of a regularization parameter. These problems differ from those more commonly encountered digital photographic reconstruction, which can be solved by linear programming, because the correct 1 -norm of a complex vector is not the sum of absolute values of real numbers but the sum of moduli of complex numbers. Solutions that enforce sparsity in the magnitude image are ideally suited to the radar imaging problem. This is because the desired image is, by construction, the locus of stationary phase points. This is always a subspace of low dimension embedded in the path space q. A typical example of this embedded subspace of stationary phase points is the isodoppler and isorange contours in a SAR image, which intersect at points in the image plane. Another is the so-called skin-return from an extended target in Inverse Synthetic Aperture (ISAR). In the three-dimensional q-space of imaging, this skin return is a two-dimensional sheet embedded in a volume. Generalized compressive processing is thus one of a family of superresolution techniques that use algebraic methods to eliminate sidelobes and estimate the underlying target parameters as closely as possible in the presence of noise. It is possible to recast the complex 1 minimization problem exactly as a Second Order Cone Program (SOCP) [4]. SOCP is a more general type of convex optimization than linear programming, and is capable of solving a very wide variety of convex optimization problems. A fairly exhaustive list of SOCP-solvers includes native MATLAB solvers SeDuMi [5] and SDPT3 [6], commercial solvers Gurobi [7] and MOSEK [8], and three new embedded solvers written in C: Embedded Cone Solver (ECOS) [9], Primal Dual Operator Splitting solver (PDOS) [10], and Splitting Cone Solver (SCS) [11]. The last three solvers are small and fast codes designed specifically for embedded application by the Stanford convex optimization group. Our recent studies have focused on SCS,

978-1-4673-9286-0/15/$31.00 ©2015 IEEE

III. R ADAR A PPLICATIONS OF SOCP

A. Array Beamsteering We first consider a uniform linear array of reciever elements employed to locate sources in the far field of the radar sensor. This case is treated in textbooks using a different notation. It provides a concrete introduction to the ideas and symbols discussed above. A plane wave of a single frequency impinges on an array of elements spaced uniformly along a line. If we reference the phase to the first element in the line, the phase depends only on the element number, and not on frequency or time. Also, there is only one parameter in the phase model, namely the angle θ between the normal to the hypothesized plane wave and the normal to the linear array. The corresponding phase function is 2πf ∆ (e − 1) sin θ (6) c where f is the frequency, c is the speed of light, e ∈ {1 . . . NE } is the integer index of the element, starting from the reference element at the end, and ∆ is the inter-element spacing, assumed constant. The steering vector, depending on the modeled angle θ has elements φ (e|θ) = −

θ|e = e−iφ(e|θ)

(7)

Figure 1 shows the response function σ(θ) that is obtained by sweeping through the generalized coordinate θ and applying Equation 1, the signal vector |z corresponding to three sources in the far field. In this elementary case, the generalized steering vector corresponds exactly to the steering vector in its original context of conventional array beamforming, and the probability density σ(θ) over path space corresponds to the signal response of the beamformer. B. Radar Imaging Consider a simple case of ISAR imaging with a set of point scatterers in a fixed plane, rotating about a common axis at angular frequency ω. We assume that the radar signal consists of wideband pulses with Nyquist frequencies labeled by f and pulse times labeled by t. The phase model now depends on measurement indices f and t and on the generalized coordinates (r, θ, ω) according to 4πf r cos (θ + ωt) (8) c The path space now has three dimensions, two consisting of the coordinates of an image in the body fixed reference f, t|r, θ, ω =

350