FFT

FFT (Fast Fourier Transform i.e., periodogram or matched filter) is one such method, which is used in majority of the spectral estimation algorithms, and there are many fast algorithms for computing the multidimensional discrete Fourier transform. Computational Kronecker-core array algebra^[18] is a popular algorithm used as new variant of FFT algorithms for the processing in multidimensional synthetic-aperture radar (SAR) systems. This algorithm uses a study of theoretical properties of input/output data indexing sets and groups of permutations.

A branch of finite multi-dimensional linear algebra is used to identify similarities and differences among various FFT algorithm variants and to create new variants. Each multidimensional DFT computation is expressed in matrix form. The multidimensional DFT matrix, in turn, is disintegrated into a set of factors, called functional primitives, which are individually identified with an underlying software/hardware computational design.^[5]

The FFT implementation is essentially a realization of the mapping of the mathematical framework through generation of the variants and executing matrix operations. The performance of this implementation may vary from machine to machine, and the objective is to identify on which machine it performs best.^[19]

Capon method

The Capon spectral method, also called the minimum-variance method, is a multidimensional array-processing technique.^[22] It is a nonparametric covariance-based method, which uses an adaptive matched-filterbank approach and follows two main steps:

1. Passing the data through a 2D bandpass filter with varying center frequencies (${\displaystyle \omega _{1},\omega _{2}}$).
2. Estimating the power at (${\displaystyle \omega _{1},\omega _{2}}$) for all ${\displaystyle \omega _{1}\in [0,2\pi ),\omega _{2}\in [0,2\pi )}$ of interest from the filtered data.

The adaptive Capon bandpass filter is designed to minimize the power of the filter output, as well as pass the frequencies (${\displaystyle \omega _{1},\omega _{2}}$) without any attenuation, i.e., to satisfy, for each (${\displaystyle \omega _{1},\omega _{2}}$),

${\displaystyle \min _{h}h_{\omega _{1},\omega _{2}}^{*}Rh_{\omega _{1},\omega _{2}}}$ subject to ${\displaystyle h_{\omega _{1},\omega _{2}}^{*}a_{\omega _{1},\omega _{2}}=1,}$

where R is the covariance matrix, ${\displaystyle h_{\omega _{1},\omega _{2}}^{*}}$ is the complex conjugate transpose of the impulse response of the FIR filter, ${\displaystyle a_{\omega _{1},\omega _{2}}}$ is the 2D Fourier vector, defined as ${\displaystyle a_{\omega _{1},\omega _{2}}\triangleq a_{\omega _{1}}\otimes a_{\omega _{2}}}$, ${\displaystyle \otimes }$ denotes Kronecker product.^[22]

Therefore, it passes a 2D sinusoid at a given frequency without distortion while minimizing the variance of the noise of the resulting image. The purpose is to compute the spectral estimate efficiently.^[22]

Spectral estimate is given as

${\displaystyle S_{\omega _{1},\omega _{2}}={\frac {1}{a_{\omega _{1},\omega _{2}}^{*}R^{-1}a_{\omega _{1},\omega _{2}}}},}$

where R is the covariance matrix, and ${\displaystyle a_{\omega _{1},\omega _{2}}^{*}}$ is the 2D complex-conjugate transpose of the Fourier vector. The computation of this equation over all frequencies is time-consuming. It is seen that the forward–backward Capon estimator yields better estimation than the forward-only classical capon approach. The main reason behind this is that while the forward–backward Capon uses both the forward and backward data vectors to obtain the estimate of the covariance matrix, the forward-only Capon uses only the forward data vectors to estimate the covariance matrix.^[22]

APES method

The APES (amplitude and phase estimation) method is also a matched-filter-bank method, which assumes that the phase history data is a sum of 2D sinusoids in noise.

APES spectral estimator has 2-step filtering interpretation:

1. Passing data through a bank of FIR bandpass filters with varying center frequency ${\displaystyle \omega }$.
2. Obtaining the spectrum estimate for ${\displaystyle \omega \in [0,2\pi )}$ from the filtered data.^[24]

Empirically, the APES method results in wider spectral peaks than the Capon method, but more accurate spectral estimates for amplitude in SAR.^[25] In the Capon method, although the spectral peaks are narrower than the APES, the sidelobes are higher than that for the APES. As a result, the estimate for the amplitude is expected to be less accurate for the Capon method than for the APES method. The APES method requires about 1.5 times more computation than the Capon method.^[26]

Advantages

SAMV method is capable of achieving resolution higher than some established parametric methods, e.g., MUSIC, especially with highly correlated signals.

Disadvantages

Computational complexity of the SAMV method is higher due to its iterative procedure.

Eigenvector method

This subspace decomposition method separates the eigenvectors of the autocovariance matrix into those corresponding to signals and to clutter.^[5] The amplitude of the image at a point (${\displaystyle \omega _{x},\omega _{y}}$ ) is given by:

${\displaystyle {\hat {\phi }}_{EV}\left(\omega _{x},\omega _{y}\right)={\frac {1}{W^{\mathsf {H}}\left(\omega _{x},\omega _{y}\right)\left(\sum _{\text{clutter}}{\frac {1}{\lambda _{i}}}{\underline {v_{i}}}\,{\underline {v_{i}}}^{\mathsf {H}}\right)W\left(\omega _{x},\omega _{y}\right)}}}$

where ${\displaystyle {\hat {\phi }}_{EV}}$ is the amplitude of the image at a point ${\displaystyle \left(\omega _{x},\omega _{y}\right)}$ , ${\displaystyle {\underline {v_{i}}}}$is the coherency matrix and ${\displaystyle {\underline {v_{i}}}^{\mathsf {H}}}$ is the Hermitian of the coherency matrix,${\displaystyle {\frac {1}{\lambda _{i}}}}$ is the inverse of the eigenvalues of the clutter subspace, ${\displaystyle W\left(\omega _{x},\omega _{y}\right)}$ are vectors defined as^[5]

${\displaystyle W\left(\omega _{x},\omega _{y}\right)=\left[1\exp \left(-j\omega _{x}\right)\ldots \exp \left(-j(M-1)\omega _{x}\right)\right]\otimes \left[1\exp \left(-j\omega _{y}\right)\ldots \exp \left(-j(M-1)\omega _{y}\right)\right]}$

where ⊗ denotes the Kronecker product of the two vectors.

MUSIC method

MUSIC detects frequencies in a signal by performing an eigen decomposition on the covariance matrix of a data vector of the samples obtained from the samples of the received signal. When all of the eigenvectors are included in the clutter subspace (model order = 0) the EV method becomes identical to the Capon method. Thus the determination of model order is critical to operation of the EV method. The eigenvalue of the R matrix decides whether its corresponding eigenvector corresponds to the clutter or to the signal subspace.^[5]

The MUSIC method is considered to be a poor performer in SAR applications. This method uses a constant instead of the clutter subspace.^[5]

In this method, the denominator is equated to zero when a sinusoidal signal corresponding to a point in the SAR image is in alignment to one of the signal subspace eigenvectors which is the peak in image estimate. Thus this method does not accurately represent the scattering intensity at each point, but show the particular points of the image.^[5][28]

Advantages

• It is invariant to the imaging mode: which means, that it uses the same algorithm irrespective of the imaging mode present, whereas, frequency domain methods require changes depending on the mode and geometry.^[29]
• Ambiguous azimuth aliasing usually occurs when the Nyquist spatial sampling requirements are exceeded by frequencies. Unambiguous aliasing occurs in squinted geometries where the signal bandwidth does not exceed the sampling limits, but has undergone "spectral wrapping." Backprojection Algorithm does not get affected by any such kind of aliasing effects.^[29]
• It matches the space/time filter: uses the information about the imaging geometry, to produce a pixel-by-pixel varying matched filter to approximate the expected return signal. This usually yields antenna gain compensation.^[29]
• With reference to the previous advantage, the back projection algorithm compensates for the motion. This becomes an advantage at areas having low altitudes.^[29]

Disadvantages

• The computational expense is more for Backprojection algorithm as compared to other frequency domain methods.
• It requires very precise knowledge of imaging geometry.^[29]

Application: geosynchronous orbit synthetic-aperture radar (GEO-SAR)

In GEO-SAR, to focus specially on the relative moving track, the backprojection algorithm works very well. It uses the concept of Azimuth Processing in the time domain. For the satellite-ground geometry, GEO-SAR plays a significant role.^[30]

The procedure of this concept is elaborated as follows.^[30]

1. The raw data acquired is segmented or drawn into sub-apertures for simplification of speedy conduction of procedure.
2. The range of the data is then compressed, using the concept of "Matched Filtering" for every segment/sub-aperture created. It is given by-${\textstyle s(t,\tau )=\exp \left(-j\cdot {\frac {4\pi }{\lambda }}\cdot R(t)\right)\cdot \operatorname {sinc} \left(\tau -{\frac {2}{c}}\cdot R(t)\right)}$ where τ is the range time, t is the azimuthal time, λ is the wavelength, c is the speed of light.
3. Accuracy in the "Range Migration Curve" is achieved by range interpolation.
4. The pixel locations of the ground in the image is dependent on the satellite–ground geometry model. Grid-division is now done as per the azimuth time.
5. Calculations for the "slant range" (range between the antenna's phase center and the point on the ground) are done for every azimuth time using coordinate transformations.
6. Azimuth Compression is done after the previous step.
7. Step 5 and 6 are repeated for every pixel, to cover every pixel, and conduct the procedure on every sub-aperture.
8. Lastly, all the sub-apertures of the image created throughout, are superimposed onto each other and the ultimate HD image is generated.

SAR polarimetry

SAR image of Death Valley colored using polarimetry

SAR polarimetry is a technique used for deriving qualitative and quantitative physical information for land, snow and ice, ocean and urban applications based on the measurement and exploration of the polarimetric properties of man-made and natural scatterers. Terrain and land use classification is one of the most important applications of polarimetric synthetic-aperture radar (PolSAR).^[36]

SAR polarimetry uses a scattering matrix (S) to identify the scattering behavior of objects after an interaction with electromagnetic wave. The matrix is represented by a combination of horizontal and vertical polarization states of transmitted and received signals.

${\displaystyle S={\begin{bmatrix}S_{HH}&S_{HV}\\S_{VH}&S_{VV}\end{bmatrix}}}$

where, HH is for horizontal transmit and horizontal receive, VV is for vertical transmit and vertical receive, HV is for horizontal transmit and vertical receive, and VH – for vertical transmit and horizontal receive.

The first two of these polarization combinations are referred to as like-polarized (or co-polarized), because the transmit and receive polarizations are the same. The last two combinations are referred to as cross-polarized because the transmit and receive polarizations are orthogonal to one another.^[37]

Three-component scattering power model

The three-component scattering power model by Freeman and Durden^[38] is successfully used for the decomposition of a PolSAR image, applying the reflection symmetry condition using covariance matrix. The method is based on simple physical scattering mechanisms (surface scattering, double-bounce scattering, and volume scattering). The advantage of this scattering model is that it is simple and easy to implement for image processing. There are 2 major approaches for a 3${\displaystyle \times }$3 polarimetric matrix decomposition. One is the lexicographic covariance matrix approach based on physically measurable parameters,^[38] and the other is the Pauli decomposition which is a coherent decomposition matrix. It represents all the polarimetric information in a single SAR image. The polarimetric information of [S] could be represented by the combination of the intensities ${\displaystyle |S_{HH}|^{2},|S_{VV}|^{2},2|S_{HV}|^{2}}$ in a single RGB image where all the previous intensities will be coded as a color channel.^[39]

Four-component scattering power model

For PolSAR image analysis, there can be cases where reflection symmetry condition does not hold. In those cases a four-component scattering model^[36][40] can be used to decompose polarimetric synthetic-aperture radar (SAR) images. This approach deals with the non-reflection symmetric scattering case. It includes and extends the three-component decomposition method introduced by Freeman and Durden^[38] to a fourth component by adding the helix scattering power. This helix power term generally appears in complex urban area but disappears for a natural distributed scatterer.^[36]

There is also an improved method using the four-component decomposition algorithm, which was introduced for the general polSAR data image analyses. The SAR data is first filtered which is known as speckle reduction, then each pixel is decomposed by four-component model to determine the surface scattering power (${\displaystyle P_{s}}$), double-bounce scattering power (${\displaystyle P_{d}}$), volume scattering power (${\displaystyle P_{v}}$), and helix scattering power (${\displaystyle P_{c}}$).^[36] The pixels are then divided into 5 classes (surface, double-bounce, volume, helix, and mixed pixels) classified with respect to maximum powers. A mixed category is added for the pixels having two or three equal dominant scattering powers after computation. The process continues as the pixels in all these categories are divided in 20 small clutter approximately of same number of pixels and merged as desirable, this is called cluster merging. They are iteratively classified and then automatically color is delivered to each class. The summarization of this algorithm leads to an understanding that, brown colors denotes the surface scattering classes, red colors for double-bounce scattering classes, green colors for volume scattering classes, and blue colors for helix scattering classes.^[41]

Although this method is aimed for non-reflection case, it automatically includes the reflection symmetry condition, therefore in can be used as a general case. It also preserves the scattering characteristics by taking the mixed scattering category into account therefore proving to be a better algorithm.

Differential interferometry

Differential interferometry (D-InSAR) requires taking at least two images with addition of a DEM. The DEM can be either produced by GPS measurements or could be generated by interferometry as long as the time between acquisition of the image pairs is short, which guarantees minimal distortion of the image of the target surface. In principle, 3 images of the ground area with similar image acquisition geometry is often adequate for D-InSar. The principle for detecting ground movement is quite simple. One interferogram is created from the first two images; this is also called the reference interferogram or topographical interferogram. A second interferogram is created that captures topography + distortion. Subtracting the latter from the reference interferogram can reveal differential fringes, indicating movement. The described 3 image D-InSAR generation technique is called 3-pass or double-difference method.

Differential fringes which remain as fringes in the differential interferogram are a result of SAR range changes of any displaced point on the ground from one interferogram to the next. In the differential interferogram, each fringe is directly proportional to the SAR wavelength, which is about 5.6 cm for ERS and RADARSAT single phase cycle. Surface displacement away from the satellite look direction causes an increase in path (translating to phase) difference. Since the signal travels from the SAR antenna to the target and back again, the measured displacement is twice the unit of wavelength. This means in differential interferometry one fringe cycle −π to +π or one wavelength corresponds to a displacement relative to SAR antenna of only half wavelength (2.8 cm). There are various publications on measuring subsidence movement, slope stability analysis, landslide, glacier movement, etc. tooling D-InSAR. Further advancement to this technique whereby differential interferometry from satellite SAR ascending pass and descending pass can be used to estimate 3-D ground movement. Research in this area has shown accurate measurements of 3-D ground movement with accuracies comparable to GPS based measurements can be achieved.

Tomo-SAR

SAR Tomography is a subfield of a concept named as multi-baseline interferometry. It has been developed to give a 3D exposure to the imaging, which uses the beam formation concept. It can be used when the use demands a focused phase concern between the magnitude and the phase components of the SAR data, during information retrieval. One of the major advantages of Tomo-SAR is that it can separate out the parameters which get scattered, irrespective of how different their motions are.^[45] On using Tomo-SAR with differential interferometry, a new combination named "differential tomography" (Diff-Tomo) is developed.^[45]

Tomo-SAR has an application based on radar imaging, which is the depiction of Ice Volume and Forest Temporal Coherence (Temporal coherence describes the correlation between waves observed at different moments in time).^[45]

Image resolution and bandwidth

Resolution in the range dimension of the image is accomplished by creating pulses which define very short time intervals, either by emitting short pulses consisting of a carrier frequency and the necessary sidebands, all within a certain bandwidth, or by using longer "chirp pulses" in which frequency varies (often linearly) with time within that bandwidth. The differing times at which echoes return allow points at different distances to be distinguished.

Image resolution of SAR in its range coordinate (expressed in image pixels per distance unit) is mainly proportional to the radio bandwidth of whatever type of pulse is used. In the cross-range coordinate, the similar resolution is mainly proportional to the bandwidth of the Doppler shift of the signal returns within the beamwidth. Since Doppler frequency depends on the angle of the scattering point's direction from the broadside direction, the Doppler bandwidth available within the beamwidth is the same at all ranges. Hence the theoretical spatial resolution limits in both image dimensions remain constant with variation of range. However, in practice, both the errors that accumulate with data-collection time and the particular techniques used in post-processing further limit cross-range resolution at long ranges.

Image resolution and beamwidth

SAR antenna of the SAOCOM satellites.

The total signal is that from a beamwidth-sized patch of the ground. To produce a beam that is narrow in the cross-range direction, diffraction effects require that the antenna be wide in that dimension. Therefore, the distinguishing, from each other, of co-range points simply by strengths of returns that persist for as long as they are within the beam width is difficult with aircraft-carryable antennas, because their beams can have linear widths only about two orders of magnitude (hundreds of times) smaller than the range. (Spacecraft-carryable ones can do 10 or more times better.) However, if both the amplitude and the phase of returns are recorded, then the portion of that multi-target return that was scattered radially from any smaller scene element can be extracted by phase-vector correlation of the total return with the form of the return expected from each such element.

The process can be thought of as combining the series of spatially distributed observations as if all had been made simultaneously with an antenna as long as the beamwidth and focused on that particular point. The "synthetic aperture" simulated at maximum system range by this process not only is longer than the real antenna, but, in practical applications, it is much longer than the radar aircraft, and tremendously longer than the radar spacecraft.

Although some references to SARs have characterized them as "radar telescopes", their actual optical analogy is the microscope, the detail in their images being smaller than the length of the synthetic aperture. In radar-engineering terms, while the target area is in the "far field" of the illuminating antenna, it is in the "near field" of the simulated one. Careful design and operation can accomplish resolution of items smaller than a millionth of the range, for example, 30 cm at 300 km, or about one foot at nearly 200 miles (320 km).

Pulse transmission and reception

The conversion of return delay time to geometric range can be very accurate because of the natural constancy of the speed and direction of propagation of electromagnetic waves. However, for an aircraft flying through the never-uniform and never-quiescent atmosphere, the relating of pulse transmission and reception times to successive geometric positions of the antenna must be accompanied by constant adjusting of the return phases to account for sensed irregularities in the flight path. SAR's in spacecraft avoid that atmosphere problem, but still must make corrections for known antenna movements due to rotations of the spacecraft, even those that are reactions to movements of onboard machinery. Locating a SAR in a crewed space vehicle may require that the humans carefully remain motionless relative to the vehicle during data collection periods.

Returns from scatterers within the range extent of any image are spread over a matching time interval. The inter-pulse period must be long enough to allow farthest-range returns from any pulse to finish arriving before the nearest-range ones from the next pulse begin to appear, so that those do not overlap each other in time. On the other hand, the interpulse rate must be fast enough to provide sufficient samples for the desired across-range (or across-beam) resolution. When the radar is to be carried by a high-speed vehicle and is to image a large area at fine resolution, those conditions may clash, leading to what has been called SAR's ambiguity problem. The same considerations apply to "conventional" radars also, but this problem occurs significantly only when resolution is so fine as to be available only through SAR processes. Since the basis of the problem is the information-carrying capacity of the single signal-input channel provided by one antenna, the only solution is to use additional channels fed by additional antennas. The system then becomes a hybrid of a SAR and a phased array, sometimes being called a Vernier array.

Amplitude data

The amplitude information, when shown in a map-like display, gives information about ground cover in much the same way that a black-and-white photo does. Variations in processing may also be done in either vehicle-borne stations or ground stations for various purposes, so as to accentuate certain image features for detailed target-area analysis.

Phase data

Although the phase information in an image is generally not made available to a human observer of an image display device, it can be preserved numerically, and sometimes allows certain additional features of targets to be recognized.

Coherence speckle

Unfortunately, the phase differences between adjacent image picture elements ("pixels") also produce random interference effects called "coherence speckle", which is a sort of graininess with dimensions on the order of the resolution, causing the concept of resolution to take on a subtly different meaning. This effect is the same as is apparent both visually and photographically in laser-illuminated optical scenes. The scale of that random speckle structure is governed by the size of the synthetic aperture in wavelengths, and cannot be finer than the system's resolution. Speckle structure can be subdued at the expense of resolution.

Optical holography

Before rapid digital computers were available, the data processing was done using an optical holography technique. The analog radar data were recorded as a holographic interference pattern on photographic film at a scale permitting the film to preserve the signal bandwidths (for example, 1:1,000,000 for a radar using a 0.6-meter wavelength). Then light using, for example, 0.6-micrometer waves (as from a helium–neon laser) passing through the hologram could project a terrain image at a scale recordable on another film at reasonable processor focal distances of around a meter. This worked because both SAR and phased arrays are fundamentally similar to optical holography, but using microwaves instead of light waves. The "optical data-processors" developed for this radar purpose^[46][47][48] were the first effective analog optical computer systems, and were, in fact, devised before the holographic technique was fully adapted to optical imaging. Because of the different sources of range and across-range signal structures in the radar signals, optical data-processors for SAR included not only both spherical and cylindrical lenses, but sometimes conical ones.