Editing Synthetic-aperture radar (section)

=== Non-parametric methods ===

==== FFT ====

FFT (Fast Fourier Transform i.e., [[periodogram]] or [[matched filter]]) is one such method, which is used in the majority of the spectral estimation algorithms, and there are many fast algorithms for computing the multidimensional discrete Fourier transform. Computational ''Kronecker-core array algebra''<ref>{{Cite journal|last=D. Rodriguez|title=A computational Kronecker-core array algebra SAR raw data generation modeling system|journal=Signals, Systems and Computers, 2001. Conference Record of the Thirty-Fifth Asilomar Conference on Year: 2001|volume=1}}</ref> 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.<ref name=":3" />

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.<ref name=":13">{{Cite journal|last=T. Gough|first=Peter|title=A Fast Spectral Estimation Algorithm Based on the FFT|journal= IEEE Transactions on Signal Processing|volume= 42| issue =  6 |date= June 1994|pages=1317–1322|bibcode=1994ITSP...42.1317G|doi=10.1109/78.286949}}</ref>

===== Advantages =====

* Additive group-theoretic properties of multidimensional input/output indexing sets are used for the mathematical formulations, therefore, it is easier to identify mapping between computing structures and mathematical expressions, thus, better than conventional methods.<ref name=":10" />
* The language of CKA algebra helps the application developer in understanding which are the more computational efficient FFT variants thus reducing the computational effort and improve their implementation time.<ref name=":10">{{Cite journal |last1=Datcu |first1=Mihai |last2=Popescu |first2=Anca |last3=Gavat |first3=Inge |title= Complex SAR image characterization using space variant spectral analysis |journal=2008 IEEE Radar Conference |year=2008}}</ref><ref>{{Cite journal |last=J. Capo4 |title= High resolution frequency wave-number spectrum analysis |journal= Proceedings of the IEEE|volume=57 |issue= 8 |pages= 1408–1418 |date= August 1969 |doi= 10.1109/PROC.1969.7278 }}</ref>

===== Disadvantages =====
* FFT cannot separate sinusoids close in frequency. If the periodicity of the data does not match FFT, edge effects are seen.<ref name=":13" />

==== Capon method ====
The Capon spectral method, also called the minimum-variance method, is a multidimensional array-processing technique.<ref name=":5">{{Cite journal|last1=A. Jakobsson|last2=S. L. Marple|last3=P. Stoica|title=Computationally efficient two-dimensional Capon spectrum analysis|journal= IEEE Transactions on Signal Processing|volume=48|year=2000|issue=9|pages=2651–2661|doi=10.1109/78.863072|bibcode=2000ITSP...48.2651J|citeseerx=10.1.1.41.7}}</ref> It is a nonparametric covariance-based method, which uses an adaptive matched-filterbank approach and follows two main steps:
# Passing the data through a 2D bandpass filter with varying center frequencies (<math>\omega_1, \omega_2</math>).
# Estimating the power at (<math>\omega_1, \omega_2</math>) for all <math>\omega_1 \in [0, 2\pi), \omega_2 \in [0, 2\pi)</math> 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 (<math>\omega_1, \omega_2</math>) without any attenuation, i.e., to satisfy, for each (<math>\omega_1, \omega_2</math>),

: <math>\min_h h^*_{\omega_1,\omega_2} Rh_{\omega_1,\omega_2}</math> subject to <math>h^*_{\omega_1,\omega_2} a_{\omega_1,\omega_2} = 1,</math>

where ''R'' is the [[covariance matrix]], <math>h^*_{\omega_1,\omega_2}</math> is the complex conjugate transpose of the impulse response of the FIR filter, <math>a_{\omega_1,\omega_2}</math> is the 2D Fourier vector, defined as <math>a_{\omega_1,\omega_2} \triangleq a_{\omega_1} \otimes a_{\omega_2}</math>, <math>\otimes</math> denotes Kronecker product.<ref name=":5" />

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.<ref name=":5" />

''Spectral estimate'' is given as

: <math>S_{\omega_1,\omega_2} = \frac{1}{a_{\omega_1,\omega_2}^* R^{-1} a_{\omega_1,\omega_2}},</math>

where ''R'' is the covariance matrix, and <math>a^*_{\omega_1,\omega_2}</math> 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.<ref name=":5" />

===== Advantages =====
* Capon can yield more accurate spectral estimates with much lower sidelobes and narrower spectral peaks than the fast Fourier transform (FFT) method.<ref>{{Cite journal|last1=I. Yildirim|last2=N. S. Tezel|last3=I. Erer|last4=B. Yazgan|title=A comparison of non-parametric spectral estimators for SAR imaging|journal=Recent Advances in Space Technologies, 2003. RAST '03. International Conference on. Proceedings of Year: 2003}}</ref>
* Capon method can provide much better resolution.

===== Disadvantages =====
* Implementation requires computation of two intensive tasks: inversion of the covariance matrix ''R'' and multiplication by the <math>a_{\omega_1,\omega_2}</math> matrix, which has to be done for each point <math>\left(\omega_1, \omega_2\right)</math>.<ref name=":2" />

==== 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:
# Passing data through a bank of FIR bandpass filters with varying center frequency <math>\omega</math>.
# Obtaining the spectrum estimate for <math>\omega \in [0, 2\pi)</math> from the filtered data.<ref>"Iterative realization of the 2-D Capon method applied in SAR image processing", IET International Radar Conference 2015.</ref>

Empirically, the APES method results in wider spectral peaks than the Capon method, but more accurate spectral estimates for amplitude in SAR.<ref name=":6">{{Cite journal|last1=R. Alty|first1=Stephen|last2=Jakobsson|first2=Andreas|last3=G. Larsson|first3=Erik|title=Efficient implementation of the time-recursive Capon and APES spectral estimators|journal=Signal Processing Conference, 2004 12th European}}</ref> 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.<ref>{{Cite journal|last1=Li|first1=Jian|last2=P. Stoica|title=An adaptive filtering approach to spectral estimation and SAR imaging|journal= IEEE Transactions on Signal Processing|year=1996|volume=44|issue=6|pages=1469–1484|doi=10.1109/78.506612|bibcode=1996ITSP...44.1469L|doi-access=free}}</ref>

===== Advantages =====
* Filtering reduces the number of available samples, but when it is designed tactically, the increase in signal-to-noise ratio (SNR) in the filtered data will compensate this reduction, and the amplitude of a sinusoidal component with frequency <math>\omega</math> can be estimated more accurately from the filtered data than from the original signal.<ref>{{Cite journal|last1=Li|first1=Jian|last2=E. G. Larsson|last3=P. Stoica|title=Amplitude spectrum estimation for two-dimensional gapped data|journal= IEEE Transactions on Signal Processing|year=2002|volume=50|issue=6|pages=1343–1354|doi=10.1109/tsp.2002.1003059|bibcode=2002ITSP...50.1343L|doi-access=free}}</ref>

===== Disadvantages =====
* The autocovariance matrix is much larger in 2D than in 1D, therefore it is limited by memory available.<ref name=":3" />