1

Fast Fourier Methods for Synthetic

Aperture Radar Imaging

Fredrik Andersson Randolph Moses Frank Natterer

Abstract

In synthetic aperture radar one wishes to reconstruct the reflectivity function of a region on

the ground from a set of radar measurements taken at several angles. The ground reflectivity

is found by interpolating measured samples, which typically lie on a polar grid in frequency

space, to an equally spaced rectangular grid in frequency space, then computing an inverse

Fourier transform. The classical Polar Format Algorithm (PFA) is often used to perform this

interpolation. In this paper we describe two other methods for performing the interpolation and

imaging efficiently and accurately. The first is the Gridding Method, which is widely used in

the medical imaging community. The second method uses unequally spaced FFTs, a generic

tool for arbitrary sampling geometries. We present numerical and computational comparisons

of these three methods using both point scattering data and synthetic X-band radar reflectivity

predictions of a construction backhoe.

Index Terms

Synthetic aperture radar. Polar format algorithm. Unequally spaced FFT. Gridding algorithm.

Nonuniform FFT. Radar imaging. USFFT. NUFFT. NFFT.

Fredrik Andersson is with the Centre for Mathematical Sciences, Lund University, P.O. Box 118, S-22100 Lund.

fa@maths.lth.se

Randolph Moses is with the Department of Electrical and Computer Engineering, The Ohio State University, 2015

Neil Avenue Columbus, OH 43210, USA. moses.2@osu.edu

Frank Natterer is with the Institut fur Numerische und Angewandte Mathematik, Westf. Wilhelms-Universitat

Munster, Einsteinstrasse 62, D-48149 Munster, Germany. nattere@uni-muenster.de

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I. INTRODUCTION

Synthetic Aperture Radar, or SAR, is one of the most important and widely used technologies

for wide-area surveillance and mapping. SAR provides an all-weather, day-or-night capability for

imaging, and has seen significant application in several areas in which areal mapping and analysis

of ground scenes is needed. Important application areas include environmental monitoring, remote

mapping, and military surveillance [24], [9], [18], [25].

Figure 1 illustrates the basic SAR reconstruction problem. An airborne or spaceborne sensor—

which could be an aircraft, satellite, or unmanned air vehicle—traverses a path and periodically

transmits an interrogating waveform toward a region of interest. The signal is reflected back by

elements in the scene that scatter electromagnetic energy. The sensor measures the reflected signal

and processes it to remove system effects such as frequency-dependent antenna or amplifier gains,

propagation losses, and timing offsets. From these resulting signals, the goal is to reconstruct the

scene reflectivity function, f(x). The scene reflectivity may be a function of two-dimensional

location x (usually the ground plane) or three-dimensional x, depending on the application.

For many applications, the SAR measurements are taken along a linear or nearly-linear path

covering a modest angular extent, and a two-dimensional scene reflectivity function (e.g., the

“ground reflectivity” function) is sought. For these applications, the measured data as a function

of frequency and angle lies on a polar support region, denoted S. In order to efficiently convert

these measurements into a SAR image, the data are first resampled onto an equally spaced

rectangular grid and then transformed into the image domain by a fast Fourier transform. The

so-called Polar Format Algorithm (PFA) is a commonly-used approach that performs the polar-to-

rectangular interpolation in two one-dimensional steps. While computationally efficient, the PFA

algorithm applies only for relatively limited angular data; for example, it is unable to perform

the interpolation if the angular extent of the data exceeds 180◦.

In this paper we present two alternative techniques for interpolation and imaging that provide

better computational and/or numerical properties. The first is called the Gridding Method (GM),

and is a commonly-used approach in the medical imaging community (see, e.g., [16]). The

algorithm is based on a quadrature approximation to the polar inverse Fourier transform, and

allows for user-controllable reconstruction error bounds, with lower error provided at the expense

of higher computational cost. The second is based on recent developments in fast algorithms for

April 4, 2010 DRAFT

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unequally-spaced Fast Fourier Transforms (FFTs), or USFFTs, and provides a direct algorithm

with user-controlled error bounds. The two methods have historically been derived differently,

both for their applications and in terms of error estimates. However, their implementations share

several ideas. Both are based on the idea to smear (convolve) out the (unequal) samples in the

frequency domain with a smoothing kernel, and then compensate for this with a multiplicative

factor in the spatial domain. Both methods can be shown to have exponentially-decreasing

interpolation errors with linearly increasing computation in each spatial dimension, and both

provide user-selectable parameters to reconstruct the desired response to arbitrary user-selected

precision.

An outline of the remainder of this paper is as follows. In Section II we review SAR imaging

and provide a mathematical problem statement for the SAR image reconstruction problem. In

Section III we review the commonly-used Polar Format Algorithm (PFA) for SAR Imaging. In

Section IV we describe the Gridding Method. Section V presents a discrete reconstruction for-

mulation of this problem using unequally-spaced Fast Fourier Transforms (FFTs) in the discrete-

problem setting to form SAR reconstructions. In Section VI we present numerical examples

that compare and contrast the three procedures in terms of implicit assumptions, error, and

computation. Section VII presents conclusions.

NOTATION

x 2D or 3D spatial location: x = [x1, x2]T or x = [x1, x2, x3]T (m)

ω temporal frequency (rad/sec)

ξ 2D or 3D spatial frequency vector (rad/m)

j j =√−1

f Fourier transform of f in nD: f(ξ) =∫

e−jx·ξf(x) dx

ε reconstruction error tolerance

II. THE SAR IMAGE RECONSTRUCTION PROBLEM

As shown in Figure 1, a SAR data collection is obtained by measuring reflected radar wave-

forms at a set of points along a path. At each point the radar transmitted waveform is a pulse with

energy in a frequency band ω ∈ [ωmin, ωmax] (rad/s). The signal propagates toward the scene at

velocity c (m/s), and is reflected back by elements in the scene that scatter electromagnetic energy.

The sensor measures the reflected waveform. The measured return signal is processed to remove

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••

••

•

••

••

•

f(x)

scene center

Fig. 1. Synthetic Aperture Radar Problem. Reflectivity measurements are taken at several aspect angles ϕk and used

to reconstruct the scene reflectivity function f(x).

system effects such as frequency-dependent antenna or amplifier gains, propagation losses, and

timing offsets. The resulting signal in the frequency domain is denoted η(ω; β), measured at

angle β = [ϕ, θ] where ϕ is the azimuth and θ is the elevation of the radar with respect to scene

center, and where ω ∈ [ωmin, ωmax]. In practice, the continuous frequency variable ω is sampled

as ωk for k = 0, . . . , K, and measurements are taken at sampled values of β� for � = −L, . . . , L.

The general 3D SAR reconstruction problem can thus be stated as follows:

Reconstruct the function f(x), where x = [x1, x2, x3]T , from the available set of

measurements {η(ωk; β�)}k,�.

The measured signals relate to the reflectivity function f(x) in the following way. For fixed β

let η(t; β) be the 1-D inverse Fourier transform of η(ω; β):

η(t; β) = F−1[η(ω; β)] =12π

∫η(ω; β)ejωtdω (1)

The range profile η(r; β) at angle β, where r is the distance, or range, away from the radar is

given by the scaling of the time axis of η(t;β) by r ↔ ct/2, along with a translation such that

r = 0 corresponds to the scene center.

Let f(x) : R3 → C be the 3D reflectivity function in the region surrounding the scene center.

Then we have

η(t; β) =∫

f(x) δ

(ct

2− x · u(β)

)dx (2)

u(β) = [cos ϕ cos θ, sinϕ cos θ, sin θ]T

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where δ(·) is the Dirac impulse function.

The SAR reconstruction problem is a classic tomographic reconstruction problem, and can

be solved using the Projection-Slice Theorem. From equation (2) it follows that the Fourier

transforms of η and f are related by (see, e.g., [18])

η(ω; β) = f(ξ)∣∣∣ξ= 2ω

cu(β)

(3)

where f(ξ) is the Fourier transform of f(x). Equation (3) states that η(ω;β) is the Fourier

transform f(ξ) defined along the radial line at polar angle β.

2D Reconstruction

In many cases, the radar measurement are taken for a set of angles that lie on a linear or nearly-

linear path whose elevation is (nearly) constant, so measurements η(ω, β) lie on (or nearly on) a

2D surface in 3D frequency space. In such a case, reconstruction of a 3D reflectivity function f(x)

is ill-posed, and one instead seeks a 2D reconstruction f(x) where now x = [x1, x2]T represents

a plane in the 3D space. Commonly-used planes include the slant-plane, defined as the plane

passing through both the flight path and scene center (see Figure 1) and the ground plane, but

reconstructions on other planes can also be used [18]. For scattering in the scene from points in

3D that do not lie on the imaging plane, the scattered response appears in the reconstructed image

at [x1, x2] locations defined by layover projections (see, e.g., [18]). For clarity of presentation,

in this paper we will consider reconstructions on the ground plane and assume that all scattering

responses lie only on the ground plane. We thus assume that f(x) has finite support on the region

D = {x = [x1, x2]T : |x1| ≤ R1; |x2| ≤ R2} (4)

This finite-support assumption follows from the fact that radars illuminate a finite region on the

ground, owing to the limited beamwidth of its transmitting or receiving antennas (see Figure 1).

For this planar case, then, the SAR imaging problem can be formulated as a planar reconstruc-

tion problem in which we reconstruct f(x) for x = [x1, x2]T from measurements η(ω; β).

The basic procedure for 2D SAR image reconstruction is as follows. The frequency mea-

surements η(ω;β) are first projected orthogonally onto the zero-elevation frequency plane such

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that

f(ξ) = η(ω, β); β = [ϕ, θ]T (5)

ξ = [2ω

ccos θ︸ ︷︷ ︸ν

cos ϕ,2ω

ccos θ︸ ︷︷ ︸ν

sinϕ]T

= [ν cos ϕ, ν sin ϕ]T (6)

where ξ is now the 2D spatial frequency variable corresponding to x = [x1, x2]T . The projected

measurements lie on the 2D frequency plane at polar coordinates (v, ϕ); see Figure 2. After

projection, measurements f(ξ) have support on the polar sector

S =

⎧⎨⎩⎡⎣ ν cos ϕ

ν sin ϕ

⎤⎦ : νmin ≤ ν ≤ νmax, |ϕ| ≤ Φ

⎫⎬⎭ (7)

where νmin = 2ωmin cos θc and νmax = 2ωmax cos θ

c . A discrete, equally-spaced sampling in ω and β thus

yields a discrete 2D polar sampling in the frequency domain at grid points ξpk,� (see Figure 2):

ξpk,� =

⎡⎣ νk cos ϕ�

νk sin ϕ�

⎤⎦ (8)

νk = νmin + kΔν, k = 0, . . . , K

ϕ� = �Δϕ, � = −L, . . . , L

where Δν = (νmax − νmin)/K and Δϕ = Φ/L.

In the inversion process, f(ξ) is often multiplied by a window function, or apodizing filter,

V (ξ) that has support on S; V is used to suppress sidelobe artifacts in the reconstruction that

arise from the sharp transitions of nonzero data in S to zero data outside of S. Finally, the

reconstructed reflectivity function is found as the inverse Fourier transform of f cV

�= f V :

f cV (x) =

14π2

∫S

ejx·ξ f(ξ)V (ξ)︸ ︷︷ ︸�=fc

V (ξ)

dξ, (9)

The superscript c on f cV denotes that f c is a function of continuous-valued x. In what follows,

we will consider the reconstruction of f cV (x), and we will assume the support of f c

V (x) is given

by D in (4).

The above formulation implicitly includes two assumptions:

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Fig. 2. Two-dimensional SAR measurement support. The shaded sector is the support region S, and shown are grid

points where f is measured.

• The far-field assumption, in which constant phase contours of the transmitted waveform are

assumed to be lines; this is embodied in the linear term(

ct2 − x · u(β)

)that appears in (2).

• The isotropic response assumption, in which it is implicitly assumed that the backscatter

response amplitude of an element in the scene f(x) is independent of the radar interrogation

direction β. While not strictly true, it is a reasonable approximation in most cases e.g., for

narrow-angle SAR [27]. If this assumption does not hold, then reconstructed image is an

average of the angle-dependent responses over the interrogated set of angles (see, e.g., [17]).

We note that if the isotropic response assumption is not valid then (9) will take on a more

complicated form, and the inverse Fourier transform will then be replaced by a Fourier integral

operator. The practical effect will be that the point spread function will be spatially varying,

cf. [27]. Recent developments in fast computations for Fourier integral operators include [28].

These rely on the usage of USFFT algorithms similar to the ones discussed in this paper.

Two main problems appear in the reconstruction problem defined by (9). First, since the

measurements do not cover the complete frequency domain, but only the subset S, the resulting

reconstruction f cV from (9) will possess convolution artifacts, since

f cV (x) = f(x) ∗ V (x). (10)

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where ∗ denotes convolution and where V (x) is the inverse Fourier transform of V (ξ). Secondly,

since the data in practice will be sampled in the frequency domain, the integral in (9) can be

approximated as

f cV (x) ≈ fV (x), (11)

where

fV (x)�=∑

s

qsejx·ξs f(ξs)V (ξs). (12)

In (12), qs are quadrature weights, given, e.g., by qs = Δξ1Δξ2

4π2 , when the frequency domain

is sampled on an equally spaced Cartesian grid, ξs = (s1Δξ1, s2Δξ2). The primary sampling

scheme of concern in this paper is polar sampling, i.e., the sample points ξs = ξs(k,�) can

be re-parameterized on a polar grid ξpk,�, cf. (8). In this case, the quadrature weights naturally

incorporate a radial factor, e.g., qs(k,�) = νkΔνΔϕ4π2 . However, the results in this paper apply to

other sampling grids as well.

If fV (ξ) were known on an equally-spaced Cartesian grid (as above), the sum above could

be efficiently evaluated using a 2D inverse Fast Fourier Transform (FFT), which would yield an

evaluation on a Cartesian grid in the space variable x. Instead, fV is typically known on the polar

coordinate grid depicted in Figure 2. An evaluation of fV on a spatial Cartesian grid could of

course be done explicitly (without using FFT) by simply evaluating the sum in (12) for each grid

point in the spatial Cartesian grid. However, this would not be very computationally efficient,

and in many cases is prohibitively slow.

Although the radar measurements are typically not taken on a Cartesian grid, it is often

desirable to present reconstructions by computing of (12) on a Cartesian grid in x. Hence,

we choose sampling resolution parameters (Δx1,Δx2) along with integers M1,M2, and define

m = [m1, m2]T and

xm =

⎡⎣ (m1 − M1/2)Δx1

(m2 − M2/2)Δx2

⎤⎦ , 0 ≤ m1 ≤ M1 − 1, 0 ≤ m2 ≤ M2 − 1. (13)

The spatial grid thus covers the region D in (4), where R1 = M12 Δx1 and R2 = M2

2 Δx2.

In considering 2D reconstructions from measured radar data, reconstruction errors arise in a

variety of ways. First, there are errors resulting from the finite support of radar measurements,

so that one obtains the convolution with V (x) in (10), called the point-spread function of the

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imaging process. This results in a reconstruction of f cV (x) instead of f(x). Second, there are

errors resulting from the quadrature approximation in (12). Third, there may be errors in the

planar reconstruction of the 3D scene, especially when the measurement surface is not exactly

planar. Finally, radar measurements are almost always corrupted by measurement noise, and there

is corresponding noise in the reconstruction. The primary focus of this paper is the development

of efficient and accurate ways to (approximately) evaluate (12) at the grid points xm.

III. POLAR FORMAT ALGORITHM

The Polar Format Algorithm (PFA) is a classic method that is widely used for forming radar

images from SAR measurements [18]. Here, we summarize PFA image formation, following the

treatment in [18].

Consider the case in which measurements of fV (ξ) are available on the 2D polar grid points

ξpk,� according to (8). The main idea behind PFA is to interpolate the data onto a Cartesian grid

where the standard FFT can be used to rapidly evaluate the Fourier sums.

Accompanying the spatial points xm are the equally spaced grid points ξrm, defined as

ξrm = ξr

m1,m2=

⎡⎣ ξ1,m1

ξ2,m2

⎤⎦ (14)

ξ1,m1 = ξ1,min + m1Δξ1, 0 ≤ m1 ≤ M1 − 1

ξ2,m2 = ξ2,min + m2Δξ2, 0 ≤ m2 ≤ M2 − 1

where

Δξ1 =2π

M1Δx1, Δξ2 =

2π

M2Δx2.

The PFA decomposes the 2D interpolation problem into two sets of 1D interpolations. The

idea is to first interpolate in the radial direction so that each radial line is resampled to points

that lie on lines of constant and equally-spaced ξ1 values (see Figure 3(a)). Since each radial

line represents the response of a finite-support range profile that has been sampled above the

Nyquist rate, perfect interpolation of each signal is theoretically possible (neglecting the effects

of a finite sample support in the frequency domain). Specifically, the first interpolation step is

for each fixed −L ≤ � ≤ L:

• Let g�(ξ1,k) = fV (ξpk,�), 0 ≤ k ≤ K, and interpolate to

g�

(ξ1,m1cos ϕ�

), 0 ≤ m1 ≤ M1 − 1.

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ξ1

ξ2

(a) First Interpolation

ξ1

ξ2

(b) Second Interpolation

Fig. 3. Polar-to-rectangular interpolation steps in the PFA algorithm. Each radial set of samples is (a) first interpolated

onto a set of constant, equally-spaced ξ1 lines, and then (b) the nonuniformly-spaced samples along each constant-ξ1

is interpolated to a uniform ξ2 grid (bottom). Each interpolation step is one-dimensional.

• This yields fV (νm1,�) where νm1,� = ξ1,m1cos ϕ�

(cos ϕ�, sin ϕ�) = ξ1,m1(1, tanϕ�).

The second step of the PFA involves resampling the unequally-spaced samples along each

constant-ξ1 line so that the resampled points lie on constant, equally-spaced ξ2 values (see

Figure 3(b). Hence, for each fixed 0 ≤ m1 ≤ M1 − 1:

• Let hm1(ξ1,m1 tan ϕ�) = fV (νm1,�) and interpolate to hm1(ξ2,m2), 0 ≤ m2 ≤ M2 − 1.

• This yields fV (ξrm) = hm1(ξ2,m2).

The second step requires that Φ < π/2, and is generally used for cases in which Φ π/2,

so the method is designed for relatively narrow-angle polar data, and cannot be directly applied

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for aspect measurement extents ≥ 180◦.

Once the data are interpolated to the equally spaced grid, i.e., once fV (ξrn) is constructed, the

final SAR reconstruction is obtained from

fV (xm) =1

4π2

∑n

fV (ξrn)ejxm·ξr

n

=1

4π2

∑n

fV (ξrn)ej((ξ1,min+n1Δξ1)x1,m+(ξ2,min+n2Δξ2)x2,m)

= ejξ1,minm1Δx1ejξ2,minm2Δx2︸ ︷︷ ︸�=αm

14π2

∑n

fV (ξrn)ej(Δξ1Δx1m1n1+Δξ2Δx2m2n2) (15)

where |αm| = 1. We are typically only interested in the magnitude of fV (x), and thus, by

discarding the phase factor αm (15) is simplified to

|fV (xm)| =1

4π2

∣∣∣∣∣∑n

fV (ξrn)ej(Δξ1Δx1m1n1+Δξ2Δx2m2n2)

∣∣∣∣∣ , (16)

which can be evaluated by means of an FFT. Zero padding of this FFT operation can be used to

generate spatial samples at a finer sampling grid (see, e.g. [20]). This is immediately seen from

(13).

In some PFA implementations, the unapodized (V ≡ 1 on the support of S) function f is first

interpolated, then an apodizing function is applied after polar-to-rectangular conversion; see [18].

Table I summarizes the PFA algorithm.

There are a number of ways to implement the interpolation operations. One can, for example,

use nearest neighbor or linear interpolation, but these can lead to significant interpolation errors

and corresponding reconstruction artifacts, especially when Δω or Δϕ are near their correspond-

ing Nyquist sampling limits. Higher-order interpolations (e.g. a cubic spline interpolation or

truncated sinc-interpolation) results in better reconstruction accuracy (with respect to (12)), at

the expense of increased computation.

Note that one can readily quantify the quadrature accuracy in (12), but the sampling errors in

(11) are less well-defined. However, as we will show in section V-C we can quantify the error

in (11) in terms of a Toeplitz kernel (point spread function).

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TABLE I

THE POLAR FORMAT ALGORITHM (PFA) FOR POLAR-TO-RECTANGULAR INTERPOLATION AND SAR IMAGING.

Data:

{fV (ξpk,�)} for 0 ≤ k ≤ K and −L ≤ � ≤ L, where ξp

k,� is the uniform polar grid given by (8).

Parameters:

Grid size (M1, M2) and spacing: ξ1,min, ξ2,min, Δξ1, and Δξ2.

Apodizing window V and 1-D interpolation method(s).

For each � = −L, . . . , L:

Interpolate the K + 1 points

g�(νk) = fV

⎛⎝νk

⎡⎣ cos ϕ�

sin ϕ�

⎤⎦⎞⎠ for νk = νmin + kΔν, k = 0, . . . , K ,

to the M1 points g�(νm1), νm1 = ξ1,m1/(cos ϕ�) for 0 ≤ m1 ≤ M1 − 1.

For m1 = 0, . . . , M1 − 1:

Interpolate the 2L + 1 points h(α�) = f([ξ1,m1 , α�]T ) where α� = ξ1,m1 tan(ϕ�) to the M2 points

h(ξ2,m2), 0 ≤ m2 ≤ M2 − 1.

Image:

Compute fV (xm) = F−1{fV (ξr)} using a 2D inverse FFT.

IV. THE GRIDDING METHOD

The Gridding Method (GM) is a classical method for reconstructing fV (x) by directly ap-

proximating the inverse Fourier transform integral in equation (9). The method is widely-used

for medical image reconstruction [16]. The approach is usually more computationally expensive

than the Polar Format Algorithm; however, it provides two useful benefits. First, unlike PFA the

algorithm imposes no restriction on the azimuth width Φ, and in fact works even for Φ = 180◦

(i.e., when measurements cover the full circle). Second, the algorithm includes direct user control

of reconstruction error; reconstruction error can be controlled at modest increase in computational

cost, owing to the exponential error reduction with a quadratic increase in computation for 2D

reconstructions (and a cubic increase for 3D reconstructions).

The essential feature of the gridding method is the use of a weight function W (x) which has

support on

DW ={

[x1, x2]T : |x1| ≤ R1, |x2| ≤ R2

}(17)

and whose Fourier transform W (ξ) is concentrated as much as possible around ξ = 0. The

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weight function is different from the apodization function V , and unlike apodization, its effect

is removed during the processing. With W being such a weight function we have

WfV (ξ) = (W ∗ fV )(ξ) =∫S

W (ξ − η)fV (η)dη (18)

=

νmax∫νmin

ν

Φ∫−Φ

W (ξ − νu)fV (νu) dωdϕ (19)

where ∗ denotes convolution and u = [cos ϕ, sin ϕ]T .

The idea is to reconstruct fV by evaluating WfV on a Cartesian grid, computing WfV by an

inverse FFT, and then removing W .

Proper discretization of (19) is important for this method to be successful. By the trapezoidal

rule,

WfV (ξ) ≈ ΔϕΔνK∑

k=0

νk

L∑�=−L

W (ξ − ξpk,�)fV (ξp

k,�). (20)

with ξpk,� defined by (8).

We will (a) compute (20) on a rectilinear set ξ, (b) take the IFFT to arrive at WfV (x) on a

corresponding rectilinear grid of x, and (c) divide by W (x) to get fV (x). The key point in the

evaluation of (20) is that W decays rapidly. This means that only those k, � values for which

ξpk,� is close to ξ contribute significantly to the overall sum. Thus, only a few terms in the sum

have to be evaluated.

Since fV (x) = 0 outside D (see (4)), the function ν → fV (νu�) has bandwidth of at most

2π/(2R) where R =√

R21 + R2

2. Similarly, since W (x) = 0 outside DW , the function ν →W (νu�) has bandwidth of at most 2π/(2R) where R =

√R2

1 + R22. Hence the function ν →

νW (ξ − νu�)f(νu�) has bandwidth of at most 2π/(2(R + R)), suggesting that a step size

Δν ≤ π/(R + R) is sufficient to achieve a Nyquist sampling rate of this product. Note that

this sampling rate is (somewhat) higher than the Nyquist rate for fV alone; the denser sampling

provides for user-selectable reconstruction accuracy, however. The above analysis ignores that the

integral in (19) is over [νmin, νmax] rather than (−∞,+∞). However, with the apodizing function

V chosen sufficiently smooth, this effect can be considered to be small.

A reasonable choice for W is W (x) = W1(x1)W2(x2) where Wi(xi) is the Kaiser-Bessel

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window:

Wi(xi) =

⎧⎨⎩ I0(A

√(R2

i − x2i ), |xi| ≤ Ri

0, otherwise(21)

with i ∈ {1, 2}, I0 being the modified Bessel function of order zero (see e.g. [1]), and where A

is a user-selected parameter. Other good window choices include Gaussians and B-splines. The

Fourier transform of the window in equation (21) is given by

Wi(ξi) = 2sinh(Ri

√A2 − ξ2

i

)√

A2 − ξ2i

, (22)

see [19]; moreover, we have for |ξi| ≥ A.∣∣∣∣∣Wi(ξi)Wi(0)

∣∣∣∣∣ ≤ RiA

sinh(RiA)(23)

This is negligibly small for modest values of RiA. A good choice for the parameters in the

gridding algorithm is

Ri = (1 + η)Ri, RiA = 9π/2, Δξi = π/Ri

The first two conditions imply oversampling in frequency by a factor of 1 + η; η = 1 is a good

choice. The third condition gives an error tolerance from equation (22) of about 2 · 10−5 for

|ξi| ≥ 9π2Ri

. Increasing A means more accuracy, at the expense of increased computation time.

The Gridding Method algorithm is summarized in Table II.

V. UNEQUALLY-SPACED FFT (USFFT) BASED-RECONSTRUCTION

The USFFT-based reconstruction approach uses recent results in direct reconstruction from

a discrete sampling of a signal to a different discrete sampling. The approach applies for ar-

bitrary sampling geometries, not just for polar-to-rectangular sampling; however, computational

efficiency is afforded owing to the use of fast USFFTs for the regular Cartesian sampling in the

reconstruction.

The approach bears similarities to the Gridding Method. The main idea is to split the application

into two parts: one that is applied in frequency and one that is applied in space. In our case, this

amounts to “smearing out” (convolving) the unequally-spaced data onto an equally-spaced grid

by convolving it with a weight function (W ). The obtained equally-spaced data are then Fourier

April 4, 2010 DRAFT

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TABLE II

THE GRIDDING METHOD FOR NONUNIFORM INTERPOLATION AND IMAGING.

Data:

{fV (ξpk,�)} for 0 ≤ k ≤ K and −L ≤ � ≤ L, where ξp

k,� is the uniform polar grid given by (8).

Parameters:

Grid size and spacing: ξ1,min, ξ2,min, Δξ1, and Δξ2.

Apodixing window V (x), and window function W (x) with Fourier transform W (ξ).

Step 1: For each grid point m, where

ξrm =

⎡⎣ ξ1

ξ2

⎤⎦ =

⎡⎣ ξ1,min + m1Δξ1

ξ2,min + m2Δξ2

⎤⎦ , (24)

m = [m1, m2]T , 0 ≤ m1 ≤ M1, 0 ≤ m2 ≤ M2,

compute:

Zm(ξrm) = ΔϕΔν

∑(k,�):(ξr

m−ξpk,�

)∈N (A)

νkW (ξrm − ξp

k,�)fV (ξpk,�)

where νk is defined in (8) and where y = [y1, y2]T ∈ N(A) iff |y1| ≤ A and |y2| ≤ A.

Step 2: Compute the inverse 2D FFT of Zm to obtain Z(xm), where

xm =

⎡⎣ (x1)m1

(x2)m2

⎤⎦ , 0 ≤ m1 ≤ M1 − 1, 0 ≤ m2 ≤ M2 − 1

and where (xi)mi = Ri(−1 + 2miMi

) for i = 1, 2.

Step 3: Divide Z(xm) by W (xm) to obtain a reconstruction of fV (xm).

transformed by means of a standard inverse FFT. In the final step we remove the effect of the

convolution by dividing (essentially) by W in the spatial domain.

In contrast to the polar format algorithm, USFFT algorithms are not dependent on a particular

sampling geometry. In order to simplify the presentation, we will assume that the grid points ξs

are centered around the origin. As a translation of the frequency grid simply corresponds to a

spatial phase factor, it can easily be done separately–or ignored, as in the case of (16), where

only the amplitude of the reconstruction is sought.

A. Discrete Reconstruction Formulation

Let us consider the reconstruction problem from a discrete perspective. Let us assume that the

ground reflectivity function f(x) has compact support as in (4). We will start by assuming that we

April 4, 2010 DRAFT

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can represent f by samples f(xm) on the Cartesian grid xm in (13). As Fourier transformation

for the discrete sampling on the grid we will use discrete Fourier series1. Hence, our aim is to

reconstruct grid values f(xm) such that∑m

f(xm)e−jxm·ξs = f(ξs). (25)

We note the resemblance between (25) and (12). Let us gather the frequency samples into a

vector

g = [f(ξs)]s;

the f(xm) into a vector f, and the exponentials in (25) into the matrix

A = A(m, s) = e−jxm·ξs .

so that (25) can be written

Af = g (26)

Moreover, by gathering the product of the quadrature weights and the sampled apodizing function

in (12) into the diagonal matrix D with elements

D(s, s) = qsV (ξs),

we can write (12) in vector form as

fV = A∗Dg.

where fV = [fV (xm)]m is a vector of the apodized reconstruction values. Hence, we conclude

that by using (12) as a reconstruction we are actually reconstructing

fV = T f; T = A∗DA. (27)

The system (26) is typically underdetermined, and it is therefore natural to impose additional

constraints. The simplest such is to find

minf

‖Af − g‖2D + μ‖f‖2

2, (28)

1If the original (continuous) reflectivity function f is represented by ideal interpolation, then the Fourier series

transform above coincides with the original continuous formulation. Furthermore, the continuous case using similar

representations by means of translations of basis functions will coincide with the Fourier series representation to within

a multiplicative factor.

April 4, 2010 DRAFT

17

where ‖ · ‖D is the weighted �2 norm induced by D, and where μ is a (Tikhonov) regularization

parameter. This problem has the solution

f = (T + μI)−1A∗Dg(m) = (T + μI)−1fV . (29)

We again recognize the right hand side of (29) to be equal to (12). Solving (29) for f given fV

can be done, e.g., by means of iterative methods.

In (28) we add a penalty factor on the �2 norm of f in order to find a unique solution. In

many cases, it is more natural to impose other conditions, e.g., constraints on the support of f.

A natural way to do this is by considering the problem

minf

‖Af − g‖2D + μ‖f‖1. (30)

In [12] a simple, yet efficient iterative method for solving problems of the kind (30) is presented.

The iterative scheme is

fn+1 = Sμ

((I − T )fn + fV

), (31)

with I being the identity operator and Sμ being the soft thresholding operator that is pointwise

defined as

Sμ(f)(m) = max(|f(m)| − μ

2, 0) f(m)|f(m)| . (32)

Just as for (29), the key component for a fast algorithm is to be able to apply T in (27) in a fast

manner.

These kinds of algorithms have received significant attention during the recent years. We would

like to mention the connections to the more general Bregman iteration, cf. [26]. Similar methods

with faster convergence rate than the simple scheme above have also been developed, cf. [4]. For

a comprehensive overview of the field, we refer to [12], [26], [4] and the references therein.

In the following subsections, we will discuss how the matrixes T and A∗ can be applied rapidly.

The polar format algorithm and the gridding method discussed before are both fast methods that

approximate the application of A∗ for the special case when sampling is done on a polar grid in

2D (when ξs = ξpk,�). Below we will instead discuss the usage of unequally spaced (sometimes

called non-equidistant, non-equispaced) FFT routines for the fast application of A∗ and T . The

formulation is independent of the sampling geometry, and also is easily extended to the 3D case.

April 4, 2010 DRAFT

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B. Unequally-Spaced FFT

Recently developed algorithms for unequally spaced FFT algorithms [5], [13] (denoted “unequally-

spaced FFT” or USFFT in [5] and “nonuniform FFT” or NFFT in [13]) allow rapid application

of the matrices A and A∗ in the previous section, at an arbitrary and user-selectable precision.

Hence, problems in which data are either given or are sought on an equally spaced grid (either

in space or in the Fourier domain) can be rapidly computed. It is shown in [5] and [13] that

time to apply A and A∗ is O(log ε(N log N)), where ε is the desired level of accuracy. Other

references in this field include [15], [21] (these use the term “non-uniform FFT” or NUFFT). In

our case, frequency data is available at unequally spaced points and sought on an equally spaced

grid in space (application of A∗).

Using the notation h = Dg, we seek a rapid way of computing the reconstruction A∗h at the

Cartesian grid points xm defined by (13). We are thus considering the problem of computing

the matrix vector product. To reconnect to the derivation of the gridding method, let

h(ξ) =∑

s

h(s)δ(ξ − ξs), (33)

where δ denotes the Dirac delta-function.

An inverse Fourier transform yields,

h(x) =1

4π2

∑s

h(s)ejξs·x

and in the case where h is evaluated at xm,

h (xm) =1

4π2

∑s

h(s)ejξs·xm = (A∗h)(m). (34)

Now, we apply (18) to (33). Hence,

h (xm) W (xm) =1

4π2

∫ ∑s

h(s)W (ξ − ξs)ejξ·xm dξ. (35)

Similarly to the Gridding Method, it is shown in [5] that when W is chosen to be a B-spline, and

in [13], [2] when W is chosen to be Gaussian, that it is possible to approximate h(x) to arbitrary

precision, by discretizing h (x) by a trapezoidal sum. It is important that proper oversampling is

April 4, 2010 DRAFT

19

used. We define the oversampled (by η, η > 1) grid points as

ξηn =[(

n1

η− M1

2

)Δξ1,

(n2

η− M2

2

)Δξ2

]T, (36)

n = [n1, n2]T

0 ≤ n1 ≤ η(M1 − 1), 0 ≤ n2 ≤ η(M2 − 1),

in the Fourier domain. Typically, η = 2.

Loosely speaking, if the grid points ξs are contained in a region that is slightly smaller than

the ones spanned by ξηm (cf. [5] for details), then for each ε > 0 there is a function Wε such

that the error of the approximation

h(xm) ≈ Δξ1Δξ2

4η2π2√∑

p Wε(xm − ηp · (R1, R2))2(37)

×∑n

∑s

h(s)Wε(ξηn − ξs)e

jξηn·xm .

is less than ε times a factor close to one. Now, just as in the Gridding Method, the function Wε

will typically decay very quickly, and in addition, for typical choices of parameters,√∑p

Wε(xm − ηp · (R1, R2))2 ≈ Wε(xm),

0 ≤ m1 ≤ M1 − 1, 0 ≤ m2 ≤ M2 − 1.

Let us now consider the inner sum in (37),∑s

h(s)Wε(ξηn − ξs). (38)

As in the Gridding case, the function Wε should decay very rapidly to zero. We can therefore

truncate (38) and discard the terms where Wε are small and still get a good approximation to

(35). Hence, let

Z(n) =Δξ1Δξ2

4π2η2

∑{s:|Wε(ξ

ηn−ξs))|<ε}

h(s)W (ξηn − ξs). (39)

The sum (39) will in practice contain many fewer elements than that of (38), and will therefore

be much faster to compute. The missing terms will all have a factor of ε or smaller, and the error

caused by replacing (38) with (39) will be on the order of ε.

April 4, 2010 DRAFT

20

Hence, we may accurately evaluate h(xm) by using

h(xm) ≈ 1Wε

∑n

Z(n)ejξηn·xm

=φ(xm)

Wε

∑n

Z(n)e2πj(n1m1ηM1

+n2m2ηM2

),

with |φ(xm)| = 1 being a translation-induced phase factor, and where the sum above can be

rapidly evaluated using the FFT.

As in the Gridding Method, it is important that W decays rapidly. In [5] B-splines are used,

whereas [13] employs Gaussians, and [14] uses Kaiser-Bessel functions. In the case of Gaussians,

we can use

Wε(ξ) = e1

4 ln ε

((

ξ1Δξ1

)2+(ξ2

Δξ2)2),

to approximate (34) to accuracy ε.

Table III gives a summary of the USFFT-based reconstruction algorithm.

C. Artifacts in Terms of Convolutions

In this section we briefly characterize the reconstruction artifacts resulting from the USFFT-

based method, and show that the artifacts can be characterized in terms of a Toeplitz operator,

or correspondingly, a point-spread convolution kernel in the image domain.

By definition

T (m, n) = A∗DA =∑

s

ej〈ξs,m〉dke−j〈ξs,n〉.

Thus, T satisfies the Toeplitz property

T (m,n) = t(m − n), 0 ≤ mi, ni ≤ Mi − 1, (40)

where

t(m) =∑

s

qse−jξs·m, −(Mi − 1) ≤ mi ≤ Mi − 1, (41)

for i = 1, 2. The Toeplitz property of T and its connection to USFFT have been noted in [6].

The advantage of Toeplitz matrices is that they can be applied rapidly by using the FFT. To see

this, we consider,

T f(m) =∑n

T (n, m)f(n) =∑n

t(m − n)f(n). (42)

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TABLE III

2D USFFT FOR RECONSTRUCTING THE REFLECTIVITY FUNCTION.

Data:

{f(ξk,�)} for 0 ≤ k ≤ K and −L ≤ � ≤ L, where ξk,� is the uniform polar grid given by (8) (or a more

general sampling).

Parameters:

Spatial grid size and spacing: M1, M2, Δx1, and Δx2.

Apodizing window V , quadrature weights qk,�, accuracy level ε, and oversampling ratio η.

Step 1: For each grid point n compute:

Z(n) =Δξ1Δξ2

4π2η2

∑{(k,�):|Wε(ξ

ηn−ξk,�))|<ε}

qk,�V (ξk,�)f(ξk,�)W (ξηn − ξk,�)

Step 2: Compute the inverse 2D FFT of Zn to obtain:

Y (m) =∑n

Z(n)e2πj(

n1m1η(M1+1)+

n2m2η(M2+1) )

, 0 ≤ mi ≤ Mi − 1,

where the sum over n is taken over 0 ≤ ni ≤ η(Mi − 1), i = 1, 2.

Step 3: Divide Y (m) by W (xm) to obtain the approximation:

fV (xm) ≈ Y (m)

W (xm).

with fV being the sought-for reflectivity function defined in (9), to within a translation-induced phase factor.

By using f as the zero padded version of f, i.e.,

f(m) =

⎧⎨⎩ f(m), 0 ≤ mi ≤ Mi − 1, i = 1, 2

0, otherwise

for −(Mi − 1) ≤ mi ≤ Mi − 1, we note that (42) can be written as a circular convolution of t

with f, and therefore can be efficiently computed by means of an FFT (see, e.g., [20]).

Since the application of T on f in (27) corresponds to convolving f discretely with t, we refer

to t as the point spread function associated with the sample geometry (distribution of nodes ξk)

combined with the choice of apodizing filter V .

VI. NUMERICAL EXAMPLES

We present two numerical examples to illustrate the properties of the three interpolation meth-

ods presented. We first consider a synthetic point scattering model, and then present reconstruc-

April 4, 2010 DRAFT

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tion results from a high-fidelity frequency-domain radar scattering prediction of a construction

backhoe.

The first example considers a set of 13 point scatterers, nine with a peak amplitude of 0 dB

arranged in a tilted 3×3 array, along with four scatters between these, each with a peak amplitude

of -30 dB. Radar measurements are simulated for a center frequency of 9.6 GHz and a bandwidth

of 1 GHz over an azimuth range of -5◦ to 5◦. Measurements are simulated on a polar grid with

1.5 times oversampling in both the frequency and azimuth dimensions. Shown in Figure 4 are

the resulting SAR image magnitudes using the three methods described above; in the case of

PFA, both linear and cubic spline interpolation were used. In all cases, an apodization window

V (ξ) was chosen as a polar-separable function, using a Taylor window with a 100 dB sidelobe

level and n = 4. For the USFFT method, we have used a Gaussian weight function

W (ξ) = e−λ‖ξ‖22 ,

with λ = 0.357193 and K = 6. With this choice of parameters, the algorithm of Section V will

approximate the fV to an accuracy level of ε = 10−5; cf. [2]. For the Gridding method, we used

a Kaiser-Bessel weight function as described in Section IV. All reconstructions were made on

a lattice of size 1024 × 1024 pixels. We see that the USFFT, Gridding and the Polar Format

algorithm with cubic spline interpolation all produce similar reconstructions. The reconstruction

using the Polar Format Algorithm with linear interpolation is substantially worse. Note that we

have used a large amplitude range (100dB).

The bottom left panel of Figure 4 show the point spread function, i.e., the Toeplitz kernel t

defined in (41). It is clearly causing a blurring effect of the reconstructions (a-d) of Figure 4.

We briefly demonstrate the ability to remove such artifact in the bottom right panel (f). Here

we show the result after convergence of the iterative algorithm of (31), with the regulariation

parameter μ = 0.01‖fV ‖∞. We note that the point scatterers appear much sharper. A detailed

investigation of the use of �1-algorithm is beyond the scope of this paper, but we refer to [23],

[22].

As a second example, we consider the Air Force Research Laboratory (AFRL) Backhoe Data

Set [3]. This set consists of simulated wideband complex backscatter data of a construction

backhoe vehicle, given in a polar frequency-domain representation. In our experiment we use

the HH-polarization data, a frequency band of 7–13 GHz, and an azimuthal span of 110◦ in the

April 4, 2010 DRAFT

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interval (−55◦, 55◦) at zero elevation angle. All reconstructions were made on a lattice of size

1024 × 1024 pixels, corresponding to a spatial region of size [−4, 4] × [−4, 4] meters.

A Taylor apodization window was applied to the frequency domain data before reconstruction

(100 dB sidelobe level, n = 4). Figure 5 shows the image reconstruction results using the three

methods considered previously, where we have used cubic spline interpolation for the Polar

Format Algorithm. The radar center azimuth is centered at the top of the images, so the front

and left side of the backhoe is illuminated over this 110◦ azimuth interval. We see that all three

methods provide similar reconstructions with small differences; the USFFT reconstruction appears

to have the fewest artifacts of the three methods for this case. In terms of the reconstruction errors,

we can see some periodic artifacts in the polar format and the gridding reconstructions—note

the “scatterer” at approximately (−4, 0). This is not due to a multiple reflection, but rather a

wrapped-around version of the (multiple) reflection near (4, 0) resulting from the circular shift

property of the discrete Fourier transform. There are also some weak (≈ -80 db) artifacts in the

polar format reconstruction that cover the entire reconstruction region.

In this example with wide frequency range, we can clearly see multiple scattering effects. We

also see the artifacts that are due to anisotropic variations in scattering response (e.g the phases

and amplitudes of scattering vary as a function of azimuth angle). However, we note that the

majority of the artifacts originate from the data model, the point spread function and from the

approximation (11).

Table IV lists the CPU time used to perform the four reconstructions shown in Figure 4. While

the PFA method with linear interpolation is somewhat faster than the USFFT method, it gives

undesirable image artifacts. All three methods have similar time complexity. The time differences

between the methods are thus dependent on the complexity constants. These will heavily depend

on the actual implementation, for instance how much effort that is put into avoiding intrinsic

function calls and keeping cache locality. In the implementations presented here, we have put

most effort in optimizing the USFFT code in such regards. Therefore, the timings in the table

below are slightly in favor of the USFFT method. The big difference in speed for the gridding

method in the point and backhoe examples is explained by the fact that the number of (frequency)

data points is much lower in the point example than in the backhoe example. The difference in

behavior compared to the USFFT results is explained by the fact that Step 1 is more optimized

in the USFFT implementations than in the gridding implementation.

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TABLE IV

COMPUTATION TIME COMPARISON FOR THE RECONSTRUCTION METHODS. TIMES ARE IN SECONDS, USING A

STANDARD DESKTOP COMPUTER.

Method Point Example SAR Example

USFFT 0.78 1.7

PFA, linear interp 0.88 1.3

PFA, cubic spline interp 1.8 5.9

Gridding 0.93 10.2

VII. CONCLUSIONS

We have described how to reconstruct the reflectivity function of a scene from measurements

of its Fourier transform given on an unequally spaced grid, e.g., in polar coordinates. The

classical Polar Format Algorithm is often used in the radar community. It comprises two steps; a

polar-to-rectangular interpolation of frequency-domain data followed by a (fast) inverse Fourier

transform operation. The PFA method is computationally efficient, but may result in artifacts if

simple interpolation methods are used; more accurate interpolation methods, such as cubic spline

interpolation, reduce these artifacts at the expense of computation.

An alternative approach is a direct frequency-to-image domain interpolation method. The

well-known Gridding Method has been used for many years, especially in the medical imag-

ing community. This method involves direct quadrature approximation of the Fourier inversion

integral, and relies on frequency data sampled on a polar lattice. The methods of unequally

spaced FFT (USFFT) work in a similar fashion as the Gridding Method, i.e., it smears out the

(unequally spaced) data onto an equally spaced lattice, and compensates for the smearing effect

by a multiplicative factor in the spatial domain. For this procedure to work, it is crucial to work

with oversampled data. In contrast to the gridding method, the USFFT algorithm works with

arbitrary sampling points and quadrature weights. A specific choice of quadrature weights and

nodes, along with an apodizing window, give rise to a (translation invariant) point spread function,

which describes the degeneration of the reconstruction problem with insufficient data available.

Finally, while these methods are presented for the 2D reconstruction problem, they readily apply

to general d-dimensional reconstruction problems and to problems in which the measurements

April 4, 2010 DRAFT

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are on an arbitrary sampling grid (i.e. there is no requirement for uniform polar sampling).

VIII. ACKNOWLEDGEMENT

The authors would like to thank Professor Margaret Cheney for introducing this problem

jointly to them, and to the Institute for Mathematics and its Applications (IMA) for providing a

stimulating venue for initial discussions of this work.

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PFA Reconstruction − linear interpolation

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6 −100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0PFA Reconstruction − spline interpolation

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6 −100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

(a) PFA, linear interpolation (b) PFA, cubic spline interpolationGridding Reconstruction

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6 −100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0USFFT Reconstruction

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6 −100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

(c) USFFT interpolation (d) Gridding interpolation

Point spread function

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6 −100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0l1 Reconstruction

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6 −100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

(e) Point spread function (f) �1 reconstruction

Fig. 4. Reconstructions of point scattering centers by the PFA method using (a) linear and (b) cubic spline interpolation,

and by the (c) gridding and (d) USFFT methods. (e) shows the point spread function and (f) an �1 reconstruction

using (31). The simulated radar center frequency is 9.6 GHz, the bandwidth is 1 GHz, and the azimuth range is 10◦.

April 4, 2010 DRAFT

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PFA Reconstruction − linear interpolation

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(a) Backhoe Facet Model (b) PFA, linear interpolation

PFA Reconstruction − spline interpolation

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0Gridding Reconstruction

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(c) PFA, cubic spline interpolation (d) Gridding interpolation

USFFT Reconstruction

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Point spread function

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(e) USFFT interpolation (f) Point spread function

Fig. 5. Reconstructions the backhoe vehicle shown in (a) from wideband and wide-angle synthetic scattering data.

(a) facet model of the backhoe; (b) PFA reconstruction using linear interpolation; (c) PFA reconstruction using cubic

spline interpolation; (d) Gridding reconstruction; (e) USFFT reconstruction; (f) Point spread function. The simulated

radar center frequency is 10 GHz, the bandwidth is 6 GHz, and the azimuth range is 110◦.

April 4, 2010 DRAFT