The achievable performance of Synthetic Aperture Radar (SAR) localization and imaging is limited ultimately by that of the noise-free phase history data acquisition and imaging. Noise and motion errors (not investigated here) can only degrade from this performance. The range of parameters corresponding to good performance constitute an envelope of good performance. Point Scatterer location, accuracy, resolution and focus (IPR or Impulse Response) vary across the parameters of frequency, bandwidth, stand-o§ range, number of spatial (array) and temporal samples, and linear or circular áight path and their squint angle from broadside. Practical point source localization uses an iterative process to identify point source volume cell (voxel) centers and precisely locating these within the voxel by applying a damped exponential estimate of the ideal point source. Measurements between each dimension of a grid of point sources determines any distortion of the 3D image volume and IPR determines any change in focus over the grid points in the volume. IPR measurements are made with a search for the half-amplitude point of the main lobe width of the Point Scatterer response in each dimension. The performance of SAR parameters is measured for a spherical array model of level circular áight paths and their linearized alternatives. The phase history data are modeled with both single and multiple direct rays reáecting from idealized point sources or corner reáector as well as a spherical di§erential range model. Performance di§erences due to di§erent combinations of phase history data collection and imaging techniques are cataloged.
Determining accurate location with Synthetic Aperture Radar (SAR) images is hindered not only by the limited resolution of the image but also by the non-rectangular nature of the process that produces the image, and the presence of multiple returns from di§erent height objects at the same range. Three dimensional SAR volume measurements can di§erentiate di§erent height scatterers but imaging methods are limited by the measurement scenario. Interpolation of images and volumes can normalize the collected data but can add inaccuracy to pixel and voxel impulse response (IPR) and localization estimates. Regions of best achievable accuracy deÖne the imaging algorithm performance envelope. Ideal localization occurs at the estimation bound of modeled point scatterers, and is also indicated by pixel and voxel size or IPR main lobe width. The performance envelope of SAR measurement scenarios, radar parameters, and imaging and estimation algorithms can reÖne system parameters of accurate localization. Use of analogous Circular and Linear SAR (CSAR and LSAR) scenarios enables joint trade-o§ of áight path along with imaging parameters. An ideal spherical construct underlies joint CSAR and LSAR, two and three dimensional SAR scenarios. This construct also ties CSAR and LSAR imaging to spherical ray-trace creation of phase history. For di§erent scenarios deÖned through a spherical model of parameters, localization performance of the imaging envelope, IPR and estimation accuracy are examined. Along with sample size and ideal chirp waveform sampling, the azimuth and elevation aperture is varied to include ideal and sub-sampling, and extension to achieve un-aliased images and volumes. Methods expanding the envelope are investigated.
Synthetic Aperture Radar (SAR) point scatterers are exploited in SAR imaging to detect known locations and objects, and to estimate and identify their speci fic parameters. The absolute, relative, and differential location accuracy provides the basis for the performance of these functions. Absolute localization can x the location of an image. Relative localization can estimate object dimensions for identification. And the (time) differential can identify sublimation (sinking) or vibration. Major variations in point scatterers stem from di¤erences in imaging techniques and the geometric (ight) imaging scenario. This paper looks at the gamut of imaging scenarios and the accuracy of parameters estimated from them. The basic location of a point scatterer in an image depends both on the image resolution fi xed by the imaging scenario parameters and the sharpness of the point scatterer (impulse response or point spread function). In some cases sub-pixel resolution accuracy is achievable via low-rank image scatterer localization algorithms. Achievable accuracy of both of these is bounded by the imaging scenario and the uniformity of this accuracy across an image which is governed by the imaging technique. Performance varies with waveform parameters, chirp rate, image size, and imaging range. Additionally, performance varies with the different sampling rates that can be used to attain the same image resolution. The performance of scatterer localization techniques across imaging scenarios and example uses of point scatterer localization are presented.
Synthetic Aperture Radar (SAR) images and volumes may be produced through multiple antenna or multiple pass scenarios. SAR imaging runs a gamut of techniques from convolution of the recieved data with the modeled SAR aperture impulse response to correlation of the spherical wave function with the recieved wave. The former has the advantage that it produces an orthogonal point spread function in each dimension allowing easy decomposition into exponential function model of a waves. The convolution however is only valid around the aim point for a spotlight image. Modi cations presented here extend the applicable envelope of use. All potential procedures and their sequence of operations are presented and examined for valued aspects including the amenability to decomposition and its validity. The extension from 2D SAR to 3D SAR may also be a¤orded to the production of 2D SAR techniques ranging between the convolution and correlation and intervening techniques with varying measures of success. As in other SAR data collections the aperture may be subsampled with imaging resolution and coverage implications. This range of use including applicable aperture size, sampling rate and squint are explored for 2D and 3D scenarios. 2D and 3D impulse response functions their accuracy and its extent throughout the image or volume are calculated. Example images and SAR image volumes are presented.
KEYWORDS: Synthetic aperture radar, 3D image processing, 3D modeling, Data modeling, Image filtering, Scattering, Algorithm development, Fourier transforms, Signal to noise ratio, Monte Carlo methods
The desire for insightful and automated segmentation or decomposition of 3D Synthetic Aperture Radar (SAR) imagery, and other 3D collections of complex detect electromagnetic wave field data leads to decomposition models with a basis such at the 3D complex exponential. This paper presents the Cramer Rao Bound (CRB) for estimation of the sum of 3D complex damped exponentials including their complex amplitude, and complex frequency in three dimensions. Synthetic 3D rectangular wave field and SAR data are decomposed into 3D damped exponentials to the CRB accuracy in examples. As in the 1D and 2D case the 3D case for a single exponential in rectangular coordinates can be directly related to the 3D Fourier Transform (FT) and its estimation accuracy. The use of SAR data presents several additional complexities. The use of linear flight path SAR is the most appropriate for complex damped exponentials, but is by the nature of SAR also approximate. Other SAR modalities such as circular, other curvilinear, or other non-uniformity ight paths are prevalent but present a different multidimensional Impulse Response Function (IPR) and are expected to deviate for the accuracy of the CRB derived here. Additionally, sampling, interpolation, and corrections to an idealized ight path are minimized or ignored in creating synthetic SAR data sets and in applying the 3D complex damped exponential decomposition to the data. The parameters of 3D complex damped exponentials will be estimated by several algorithms and their accuracy with be compared with the 3D complex damped exponential Cramer Rao bound.
Synthetic Aperture Radar (SAR) creates a 2-D (azimuth-range) image from radar pulses collected equally-spaced along a linear áight path. One 3-D scenerio collects these pulses at each collection point along the path from a linear (elevation) array orthogonal to the áight path. From this 3-D data set images (to a pixel accuracy) or array processing (to subpixel accuracy) allows strong scatterers to be located. Streamlined algorithms are needed for such practical image and volume reáectively function formation. Sacchini, Steedly and Moses (1993)3 presents a 2-D Total Least Squares (TLS) Prony method that robustly identiÖes 2-D scatterer locations in SAR images. In this method scatterer coordinates are matched by Ötting the data in each dimension, Ötting the resultant amplitudes in the cross-dimension and then matching the highest energy pairs in both these sets. This matching can produce excellent results for TLS Prony and for other 1-D scatterer localization algorithms. The algorithm is extended here to supply 3-D scatterer locations for simulated 3-D SAR data. Previous results for 3-D data show good localization using 2-D TLS Prony on azimuth-elevation slices and interpolating the range location between slices. Thresholding of the highest energy points, however, is required to Önd the actual location of scatterers. Range accuracy is also limited due to use of only the two closest range samples. Consistency of results is di§erent for di§erent amplitude scatterers. This paper produces results for a new 3-D TLS Prony method. Algorithm accuarcy, bias, robustness in di§erent scenarios are examined.
Integral in locating point scatterers in Synthetic Aperture Radar (SAR) data is the ability to match location
estimates in each dimension. This is due in some sense to the fact that the fundamental theorem of algebra nds
unique locations only in one dimension. In SAR images this involves at least a search of four possible combination
for two scatterers. In a set of multiple elevation SAR (3-D) images with more than one scatterer combinations
increase dramatically. The paper examines several suboptimal methods and their e¢ ciency matching scatterers
in one or more dimensions to their unique locations compared to the (un-achievable) exhaustive search. Many
heuristic methods exist in two dimension (location maxima, alternating maximization in each dimension) and
some (radar tracking) methods exist in three dimensions (Munkres, probabilistic maximization). Algorithms
range from simply selecting maximums (easy in 2D; complex in multiple images, 3D) to multidimensional con-
strained interpolations. In some algorithms the extra degrees of freedom present in two dimensional localization
are exploited to increase accuracy. These methodologies can also be extended to three dimensions. The paper
examines proposed combinations of these especially suitable to the 3-D SAR problem. Simulations with results
for di¤erent algorithms compare promising alternatives to solve this problem.
In Synthetic Aperture Radar (SAR) the resultant image gives not only the complex reflectivity of image points but also their interdependency with respect to time and observation angle. In range or fast-time changes in reflectivity are expectedly slight, however, in observation azimuth or slow-time the reflectivity pattern, movement or vibration of strong scatterers is revealed. These azimuth signals can subsequently reveal pass to pass changes over inter-pass time or observation elevation. Key to extracting the slow-time signals is the imaging method involved. If imaging preserves the phase function across azimuth then the time or aspect phenomenon riding on top of the phase can be extracted. In other cases the phase is distorted or overridden by imaging artifacts. The choice of imaging method is fundamental in determining not only image resolution but also the fidelity with which secondary signals along the aperture can be determined. The achievable envelope of secondary signal amplitude, bandwidth and resolution are determined here for several imaging methods including the fraction Fourier transform, deramping, range Doppler, chirp scaling, wave-front and matched filtering. Method of extracting these secondary azimuth dependent signals are developed and results are presented for an orbital scenario. Naturally sampling speed, pulse spacing and the flight path in slow-time enclose the largest potential envelope of measurable secondary signals while the selection of imaging method restricts the potential measurable signals to a smaller envelope. Sampling restrictions and bounds on range migration curvature for different imaging methods are also found.
The location of point scatterers in Synthetic Aperture Radar (SAR) data is exploited in several modern analyzes including persistent scatter tracking, terrain deformation, and object identification. The changes in scatterers over time (pulse-to-pulse including vibration and movement, or pass-to-pass including direct follow on, time of day, and season), can be used to draw more information about the data collection. Multiple pass and multiple antenna SAR scenarios have extended these analyzes to location in three dimensions. Either multiple passes at different elevation angles may be .own or an antenna array with an elevation baseline performs a single pass. Parametric spectral estimation in each dimension allows sub-pixel localization of point scatterers in some cases additionally exploiting the multiple samples in each cross dimension. The accuracy of parametric estimation is increased when several azimuth passes or elevations (snapshots) are summed to mitigate measurement noise. Inherent range curvature across the aperture however limits the accuracy in the range dimension to that attained from a single pulse. Unlike the stationary case where radar returns may be averaged the movement necessary to create the synthetic aperture is only approximately (to pixel level accuracy) removed to form SAR images. In parametric estimation increased accuracy is attained when two dimensions are used to jointly estimate locations. This paper involves jointly estimating azimuth and elevation to attain increased accuracy 3D location estimates. In this way the full 2D array of azimuth and elevation samples is used to obtain the maximum possible accuracy. In addition the independent dimension collection geometry requires choosing which dimension azimuth or elevation attains the highest accuracy while joint estimation increases accuracy in both dimensions. When maximum parametric estimation accuracy in azimuth is selected the standard interferometric SAR scenario results. When maximum estimation accuracy in elevation is selected the multiple baseline interferometric SAR scenario results. Use of a 2D parametric estimation method attains the best accuracy possible in both dimensions. When in some scenarios particularly the orbital case where the azimuth dimension is only approximately linear the full accuracy increase of linear joint azimuth and elevation is not fully attained. Images and point cloud estimates are shown for several linear and orbital SAR scenarios. Images provide a visual representation of the data while the quantitative point cloud data is a direct input for the multiple analyzes listed earlier.
The 3-D Fractional Fourier Transformation (FrFT) has unique applicability to multi-pass and multiple receiver
Synthetic Aperture Radar (SAR) scenarios which can collect radar returns to create volumetric reflectivity data.
The 3-D FrFT can independently compress and image radar data in each dimension for a broad set of parameters.
The 3-D FrFT can be applied at closer ranges and over more aperture sampling conditions than other imaging
algorithms. The FrFT provides optimal processing matched to the quadratic signal content in SAR (i.e. the
pulse chirp and the spherical wave-front across the aperture). The different parameters for 3-D linear, circular,
and orbital SAR case are derived and specifi…c considerations such as squint and scene extent for each scenario are
addressed. Example imaged volumes are presented for linear, circular and orbital cases. The imaged volume is
sampled in the radar coordinate system and can be transformed to a target based coordinate system. Advantages
of the FrFT which extend to the 3-D FrFT include its applicability to a wide variety of imaging condition (standoff range and aperture sub-sampling) as well as inherent phase preservation in the images formed. The FrFT closely
matches the imaging process and thus is able to focus SAR images over a large variation in standoff ranges
specifi…cally at close range. The FrFT is based on the relationship between time and frequency and thus can
create an image from an under-sampled wave-front. This ability allows the length of the synthetic aperture to
be increased for a fixed number of aperture samples.
The almost unique ability of azimuth deramping to preserve a smooth phase function in azimuth is exploited here
to link two disparate spatial processing methods, Direction of Arrival (DOA) localization and Interferometric
Synthetic Aperture Radar (IFSAR) and explore the achievable accuracy inherent in their common measurement
scenario. Deramping in range quickly provides a rst component for point source localization. Deramping
in azimuth is phase preserving and provides an approximate localization in azimuth that is more accurate
over narrower apertures and can be corrected in scenarios involving range migration and for its point source,
azimuth location dependence. In cross-track IFSAR two antenna measurements azimuth/elevation DOAs can be
calculated from their smooth azimuth functions at each range with a 1 D parametric estimate (exponential model)
of point sources. Joint frequency estimates (both antennae) provide the azimuth DOA while the phase di¤erence
between antenna amplitude estimates provides the elevation DOA. The cross track antenna measurements can
also be processed via the IFSAR methodology producing two SAR images and the phase di¤erence between
the two (an interferogram). This provides two images coordinates and a height for each pixel. The connection
between the phase history DOA localization and the IFSAR is used to attain accuracy bounds for IFSAR.
Extrapolation of the bounds is provided from two spatially un-aliased antennas to IFSAR scenarios with large
baseline separations of the antennas. In addition imaging from the azimuth-elevation-range localization data and
its ability to minimize layover (building tops imaged closer than their bases) is explored.
In synthetic-aperture radar (SAR) returned signals, ground-target vibrations introduce a phase modulation that
is linearly proportional to the vibration displacement. Such modulation, termed the micro-Doppler effect, introduces
ghost targets along the azimuth direction in reconstructed SAR images that prevents SAR from forming
focused images of the vibrating targets. Recently, a discrete fractional Fourier transform (DFrFT) based method
was developed to estimate the vibration frequencies and instantaneous vibration accelerations of the vibrating
targets from SAR returned signals. In this paper, a demodulation-based algorithm is proposed to reconstruct
focused SAR images of vibrating targets by exploiting the estimation results of the DFrFT-based vibration
estimation method. For a single-component harmonic vibration, the history of the vibration displacement is
first estimated from the estimated vibration frequency and the instantaneous vibration accelerations. Then a
reference signal whose phase is modulated by the estimated vibration displacement with a delay of 180 degree is
constructed. After that, the SAR phase history from the vibration target is multiplied by the reference signal and
the vibration-induced phase modulation is canceled. Finally, the SAR image containing the re-focused vibration
target is obtained by applying the 2-D Fourier transform to the demodulated SAR phase history. This algorithm
is applied to simulated SAR data and successfully reconstructs the SAR image containing the re-focused
vibrating target.
When vibrating objects are present in a Synthetic Aperture Radar image they induce a modulation in the
pulse-to-pulse Doppler collected. At higher frequencies (up to a sampling limit dictated by half the PRF) the
modulation is low amplitude due to physical limits of vibrating structures and swamped by the Doppler from
static objects (clutter). This paper presents an orthogonal subspace transform that separates the modulation
of a vibrating object from the static clutter. After the transformation the major frequencies of the vibration
are estimated with asymptotically (as the number of pulses increases) decreasing variance and bias. Although
the e¤ects and SAR image artifacts from vibrating objects are widely known their utility has been limited to
high signal-to-noise, low frequency vibrating objects. The method presented here lowers the minimum required
signal-to-noise ratio of the vibrating object over other methods. Additionally vibrations over the full (azimuth-
sampled) frequency range from one over the aperture time to the pulse repetition frequency (PRF) are equally
measured with respect to the noise level at each speci c frequency. After separation of the vibrating and static
object signal sub-spaces any of the many spectral estimation methods can be applied to estimate the vibration
spectrum.
In synthetic-aperture radar (SAR), ground-target vibrations introduce a phase modulation in the returned signals,
a phenomenon often referred to as the micro-Doppler effect. Earlier work has shown that the problem of
estimating common ground-target vibrations can be transformed into the problem of successively estimating
chirp parameters of the returned signal in properly sized subapertures. Recently, a method based on the discrete
fractional Fourier transform (DFRFT) was proposed, in conjunction with the subaperture framework, to estimate
target vibrations in the absence of noise. In this paper a pseudo-subspace approach is employed to extend the
applicability of the DFRFT-based vibration-estimation method to signals that are corrupted by white noise.
The new algorithm first calculates the inverse discrete Fourier transform of row and column projections of
the magnitude of the DFRFT spectrum of the SAR returned signal to obtain two vectors. Next, covariance
matrices are estimated from the sample covariance matrices of the two vectors. A pseudo-subspace approach is
then applied to the covariance matrices to yield the pseudo-spectra. The chirp rate of the signal is estimated
by finding the principle frequency component in the corresponding pseudo-spectrum. Monte-Carlo simulations
demonstrate that the proposed method generally offers improved mean-square-error performance in the presence
of noise compared to the direct DFRFT-based method.
The 2-D Fractional Fourier Transform (FRFT) has been shown to be applicable to the Synthetic Aperture
Radar (SAR) imaging problem. Streamlined versions presented here makes the 2-D FRFT comparable with and
slightly faster than the Range Doppler (RD) and Extended Chirp Scaling (ECS) methods. The 2-D FRFT is
streamlined by eliminating redundancy due to the fact that the same fractional angle is applied to each pulse in
the SAR phase history's range dimension while one other fractional angle is applied across each range-gate in the
phase history's azimuth dimension. Eliminating the redundancy and approximating the 2-D Fractional Fourier
Transform operation in each dimension produces several streamlined 2-D FRFT methods as well as a very fast
approximate 2-D FRFT. The computational order of the fast approximate 2-D FRFT is less than that of other
corrective SAR imaging techniques. Examples of SAR imaging with these streamlined and approximate FRFTs
are given as well as a comparison of the computational speed and impulse response of the full, streamlined and
approximate 2-D FRFT, and the RD and ECS methods of SAR imaging.
Recent reports on the effects of vibrating targets on synthetic-aperture radar (SAR) imagery and the potential
of SAR to extract non-stationary signatures have drawn significant interest from the remote-sensing community.
SAR returned signals are the superposition of the transmitted pulses modulated by both static and non-static
targets in both amplitude and phase. More precisely, the vibration of a target causes a small sinusoid-like frequency
modulation along the synthetic aperture (slow time), whereby the phase deviation is proportional to
the displacement of the vibrating object. By looking at successive small segments in slow time, each frequency
modulated pulse can be tracked and further approximated as a piecewise-linear frequency-modulated signal. The
discrete-time fractional Fourier transform (DFRFT) is an analysis tool geared toward such signals containing linear
frequency modulated components. Within each segment, the DFRFT transforms each frequency-modulated
component into a peak in the DFRFT plane, and the peak position corresponds to the frequency modulation rate.
A series of such measurements provides the instantaneous-acceleration history and its spectrum bears the vibrating
signature of the target. Additionally, when the chirp z-transform (CZT) is incorporated into the DFRFT,
vibration-induced modulations can be identified with high resolution. In this work, the interplay amongst SAR
system parameters, vibration parameters, the DFRFT's window size, and the CZT's zoom-in factor is characterized
analytically for the proposed SAR-vibrometry approach. Simulations verify the analysis showing that the
detection of vibration using the slow-time approach has significantly higher fidelity than that of the previously
reported fast-time approach.
In this paper, we develop a method for determining the vibration spectrum and vibrating direction of a vibrating
object measured with Synthetic Aperture Radar. The methodology presented here is performed after the
vibration history has been extracted from the SAR phase history by some other technique; then, our method is
applied. The method is tested here with simulated data to verify its performance and to determine the conditions
required for good vibration spectrum and direction estimates.
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