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Review Papers

Target detection in synthetic aperture radar imagery: a state-of-the-art survey

[+] Author Affiliations
Khalid El-Darymli

C-CORE, Captain Robert A. Bartlett Building, Morrissey Road, St. John’s, Newfoundland, A1C 3X5, Canada

Memorial University, St. John’s, Newfoundland, A1B 3X5, Canada

Peter McGuire

C-CORE, Captain Robert A. Bartlett Building, Morrissey Road, St. John’s, Newfoundland, A1C 3X5, Canada

Desmond Power

C-CORE, Captain Robert A. Bartlett Building, Morrissey Road, St. John’s, Newfoundland, A1C 3X5, Canada

Cecilia Moloney

Memorial University, St. John’s, Newfoundland, A1B 3X5, Canada

J. Appl. Remote Sens. 7(1), 071598 (Mar 18, 2013). doi:10.1117/1.JRS.7.071598
History: Received July 10, 2012; Revised November 27, 2012; Accepted November 29, 2012
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Open Access Open Access

Abstract.  Target detection is the front-end stage in any automatic target recognition system for synthetic aperture radar (SAR) imagery (SAR-ATR). The efficacy of the detector directly impacts the succeeding stages in the SAR-ATR processing chain. There are numerous methods reported in the literature for implementing the detector. We offer an umbrella under which the various research activities in the field are broadly probed and taxonomized. First, a taxonomy for the various detection methods is proposed. Second, the underlying assumptions for different implementation strategies are overviewed. Third, a tabular comparison between careful selections of representative examples is introduced. Finally, a novel discussion is presented, wherein the issues covered include suitability of SAR data models, understanding the multiplicative SAR data models, and two unique perspectives on constant false alarm rate (CFAR) detection: signal processing and pattern recognition. From a signal processing perspective, CFAR is shown to be a finite impulse response band-pass filter. From a statistical pattern recognition perspective, CFAR is shown to be a suboptimal one-class classifier: a Euclidean distance classifier and a quadratic discriminant with a missing term for one-parameter and two-parameter CFAR, respectively. We make a contribution toward enabling an objective design and implementation for target detection in SAR imagery.

Figures in this Article

Synthetic aperture radar (SAR) offers distinctive active remote sensing capabilities for both military and civilian applications. Target, clutter, and noise are three terms of military origins associated with automatic target recognition (ATR), and their definition depends on the application of interest. In the case of SAR imagery, target refers to the object(s) of interest in the imaged scene. Clutter refers to manmade (building, vehicles, etc.) and/or natural objects (trees, topological features, etc.) that tend to dominate the imaged scene. Noise refers to imperfections in the SAR image which are a result of electronic noise in the SAR sensor, as well as computational inaccuracies introduced by the SAR signal processor. The general structure of an end-to-end ATR system for SAR imagery (SAR-ATR), as reported in the literature, is depicted in Fig. 1. To account for the prohibitive amounts of processing pertaining to the input SAR imagery, the strategy is to divide and conquer. Accordingly, the SAR-ATR processing is split into three distinctive stages: detector (also known as prescreener), low-level classifier (LLC, also known as discriminator), and high-level classifier (HLC).19 The first two stages together are commonly known as the focus-of-attention module. While this is the most common structure reported in the literature, it should be highlighted that (theoretically) there is no restriction on the number of stages.

Grahic Jump LocationF1 :

General structure for an end-to-end SAR-ATR system.

As depicted in Fig. 1, the input SAR image creates an extremely high computational load due to its high resolution and/or the presence of various clutter types and objects. As the SAR data progresses throughout the SAR-ATR processing chain, its load is reduced. The HLC stage deals with SAR data that has relatively lower computational load. To the contrary, the computational complexity of the SAR-ATR chain increases as the SAR data progresses from the front-end stage toward the back-end stage.

Detection is the front-end stage in any SAR-ATR processing chain. The detector interfaces with the input SAR image to identify all regions of interest (ROIs), thus ROIs can be passed-in to the LLC stage for further analysis. One may think of the detector as a dimensionality reduction scheme that properly reduces the dimensionality of the SAR data. The detector should be designed to balance the tradeoff between computational complexity, detection efficacy, and outlier rejection. On the one hand, it is required that the detector is relatively computationally simple, thus it can operate in real-time or near-real-time. On the other hand, it is required that the detector enjoys a low probability of false alarm (PFA), and a high probability of detection (PD). Indeed, these often conflicting factors distinguish one detector from another.

There are numerous strategies for implementing the detector. This is evident in the overwhelming number of research articles published on the topic in the open literature. Different researchers tend to approach the topic from various perspectives. This makes it even more challenging and time consuming to relate the various research findings and to grasp the relationship between these various approaches. This shows a dire need for a survey that offers an umbrella under which various research activities can be broadly probed and taxonomized. This is precisely the goal of this paper.

In this paper, we restrict our attention to single-channel SAR imagery (i.e., single polarization). This is because the development of our survey is motivated by our endeavor to develop SAR-ATR algorithms for Spotlight mode Radarsat-2 data, which is a single channel.10 However, many of the topics described in this survey are either applicable or extendable to multichannel SAR imagery. For readers interested in multichannel SAR image processing, please refer to pertinent references.1117

The remainder of this paper is organized as follows. In Sec. 2, taxonomy of the detection methods is introduced. Primarily, three taxa are proposed: single-feature-based, multifeature-based, and expert-system-oriented. In Sec. 3, the various methods for implementing the detection module are comprehensively surveyed. First, a classification methodology for the various strategies under each taxon is introduced. Next, a description of the different sub-taxa is elaborated. The description commences with the Bayesian approach being the optimal approach, and it ends with the various detection strategies that fall under the expert-system-oriented taxon. Representative examples pertinent to SAR imagery are carefully chosen, and relevant comments are pinpointed throughout this section. The issues approached include proper choice of the size of the constant false alarm rate (CFAR) sliding window, CFAR as a suboptimal one-class classifier, CFAR loss, and cell averaging CFAR (CA-CFAR) as the baseline algorithm for comparison with other CFAR techniques. In Sec. 4, a compact tabular comparison between the surveyed methods is offered. In Sec. 5, a novel discussion is presented. The issues covered under the discussion include suitability of SAR data models, understanding the multiplicative SAR data models, and two unique perspectives on CFAR detection: a signal processing perspective and a statistical pattern recognition perspective. In Sec. 6, the paper is concluded.

The detection module takes the entire SAR image and identifies the ROIs. Ultimately, the detected regions in the image are passed-in to the next stage in the SAR-ATR chain for further analysis. The goodness of any detection module is typically judged based upon three aspects of significance: computational complexity, PD, and false alarm rate (also known as PFA). The detection module should enjoy a low computational complexity such that it operates in real-time or near-real-time. This is in contrast to the succeeding stages in the SAR-ATR chain, which are relatively more computationally expensive. Further, a good detection module should provide a means to refine detections, reduce clutter false alarms, and pass ROIs; thus the detection method enjoys a reasonable PFA and acceptable PD.

We broadly taxonomize the detection algorithms reported in the open literature into three major taxa: single-feature-based, multifeature-based, and expert-system-oriented. This taxonomy is depicted in Fig. 2.

Grahic Jump LocationF2 :

Major taxa for implementing the detection module.

The single-feature-based taxon bases the detection in the SAR image on a single feature; typically the brightness in the pixel intensity commonly known as the radar cross-section (RCS). Various methods in the literature fall under this taxon. The single-feature-based approach is placed at the base of the pyramid in Fig. 2 because it is the most common and widely used in the literature. Further, the single-feature-based approach is the building block for the other two taxa.

The multifeature-based taxon bases the detection decision on a fusion of two or more features extracted from the input SAR image. Besides RCS, additional features that can be inferred and fused include multiresolution RCS and fractal dimension. Obviously, this taxon builds on the previous taxon and is expected to provide relatively improved detection performance, along with fewer false alarms. Multiple methods in the literature fall under this taxon.

Finally, the expert-system-oriented taxon is the most sophisticated. It extends the two aforementioned taxa and utilizes a multistage (two or more stages) artificial intelligence (AI) approach that bases the detection process in the SAR image on exploitation of prior knowledge about the imaged scene, clutter, and/or target(s). Prior knowledge is exploited through various means such as image segmentation, scene maps, previously gathered data, etc.

As the sophistication of the detection taxon increases, the complexity-performance tradeoff arises. Caution should be taken when opting for a certain approach in order to balance this tradeoff carefully.

Based on the aforementioned taxonomy, we broadly classify the various detection schemes and relevant methods reported in the literature in Fig. 3.

Grahic Jump LocationF3 :

The three major target detection taxa and multiple sub-taxa and methods under each of them.

Primarily, under the single-feature-based taxon, the sliding window CFAR (CFAR-based) sub-taxon is the most popular. The various CFAR methods can be viewed through three perspectives. First, based on the specifications of the sliding window, there is fixed-size versus adaptive, as well as rectangle-shaped versus nonrectangle-shaped. Second, based on the strategy used to implement the CFAR technique, there are various strategies, including cell-averaging CFAR (CA-CFAR), smallest of CA-CFAR (SOCA-CFAR), greatest of CA-CFAR (GOCA-CFAR), and order statistics CFAR (OS-CFAR). Third, based on the method used to estimate the threshold (for a desired PFA) in the boundary ring and/or the approach utilized for estimating the target signature (for a desired PD), two subclasses emerge: parametric and nonparametric. Under the parametric subclass, two approaches are recognized: only background modeling and background and target modeling. A choice of the parametric model that best represents the SAR data in use has to be made among various parametric models. Unlike the parametric approach, the nonparametric approach does not assume any form for the background/target model(s). Rather, it directly infers an approximate model from the training data. One such method to perform the model inference is the kernel-density-estimation (KDE) method.

Less popular non-CFAR-based methods, such as those that rely on a coherence image, represent the other sub-taxon of single-feature-based methods. The single-feature-based taxon has the limitation that it bases the detection decision solely on RCS, and thus it can become overwhelmed in regions in the SAR image where there is heterogeneous clutter and/or a high density of targets.

Methods under the multifeature-based taxon try to circumvent this drawback by basing the detection decision on a fusion of two or more features. Obviously, this taxon can utilize a suitable method among those presented under the single-feature-based taxon and incorporate additional features besides RCS, such as multiresolution RCS analysis, fractal dimension, etc. Multiresolution methods can be either space-scale-based or space-frequency-based. Prime examples of methods that utilize space-scale features are those based on the wavelet transform, including the discrete wavelet transform (DWT), and the continuous wavelet transform (CWT). Prime examples of methods that utilize space-frequency features include linear space-frequency methods such as the Gabor transform and the S-transform, along with bilinear (also known as quadratic) space-frequency methods such as Cohen’s class distributions (Wigner distribution, Wigner-Ville distribution, pseudo-Wigner-Ville distribution, etc.).

Finally, a more robust taxon is the expert-system-oriented approach, which incorporates intelligence into the process to guide the decision making. In its simplest form, detection decisions can be guided by a structure map of the imaged scene generated from properly segmenting the SAR image. Further, methods of AI can be appropriately integrated to achieve near-optimal context utilization.

Next we review the various methods introduced above under each taxon. Further, representative examples pertaining to SAR imagery under each method are carefully chosen and presented.

Single-Feature-Based Taxon

Single-feature-based detection algorithms base their search for target detection in the SAR image on a single feature. CFAR is the most popular single-feature-based detection algorithm. Despite the many variations of CFAR under this category, they are considered single-feature-based because they base the search for ROIs on RCS alone. Indeed, as it is evident from the numerous works published in the literature, CA-CFAR is the baseline approach for target detection in SAR imagery. To understand the limitations of the single-feature-based-CFAR approach, it is important to review its underlying assumptions.

An optimal detector (theoretically speaking) should utilize the Bayesian approach, which for a zero-one cost, reduces to the maximum a posteriori (MAP) criterion18 asDisplay Formula

ΛMAP(x)=P(ωT|x)P(ωB|x)ωTωB1,(1)
where ωT is the target-class, ωB is the background or clutter-class, and P(ωT|x) and P(ωB|x) are the posteriors of the target-class and the background-class, respectively.

This is simply a binary (i.e., two-class; dichotomizer) classification problem where x is a feature vector that represents the pixel values, typically obtained from the boundary ring in a sliding window with suitable guard cells centered on the ROI. This window is typically called a CFAR stencil and is depicted in Fig. 4 for a size of 9×9pixels. The boundary ring is shown in bright green with suitable pixel labels.

Note that the choice of this stencil size here is for illustration purposes only. Proper choice of the stencil size will depend on the type of SAR image and the target size. More precisely, the size (and shape) of the guard ring will be governed by the geometrical size of the target.19 However, it should be noted that choosing the stencil dimension using only the prior knowledge of target size yields a detection loss (i.e., CFAR loss that leads to a suboptimal performance), because the backscatter of the target in SAR imagery is dependent on the operating conditions and is weakly linked with the target’s geometric shape.20 Despite these challenges, it is recommended21,22 that the target window size (i.e., pixels under test, PUTs) should be about the size of the smallest object that one wishes to detect, the guard ring window size should be about the size of the largest object, and the boundary ring window size should be large enough to estimate the local clutter statistics accurately.

Posterior probabilities can be expressed by the Bayes rule as Display Formula

P(ωT|x)=p(x|ωT)P(ωT)p(x),andP(ωB|x)=p(x|ωB)P(ωB)p(x),(2)
where p(x|ωT), and p(x|ωB) are the probability density functions (PDFs) or the likelihoods of the target-class and the background-class, respectively; P(ωT), and P(ωB) are the priors of the target-class and the background-class, respectively; And p(x) is the evidence, which is typically ignored in the MAP criterion above, because it is identical for both classes.

Accordingly, the MAP criterion can be expressed as a likelihood ratio test (LRT) as Display Formula

ΛLRT(x)=p(x|ωT)p(x|ωB)ωTωBP(ωB)P(ωT).(3)
The major problem with this expression is that, in practice, we do not have knowledge on the class priors. If we assume equal priors for the target-class and the background-class, the LRT test reduces to the maximum likelihood (ML) test. However, in practice, these two priors are not equal, and the ML test should be avoided. Typically, the Neymann-Pearson (NP) criterion is adopted where the LRT test reduces to Display Formula
ΛNP(x)=p(x|ωT)p(x|ωB)ωTωBα,(4)
where α is a detection threshold typically referred to as the threshold scaling factor for reasons that will become apparent.

Equation (4) is the main formula motivating the design for the CFAR algorithm and its variants. Indeed, many popular CFAR algorithms in the literature assume that only the background-class is characterized without characterizing the target-class, and thus they adopt a suboptimal anomaly detection (AD). This converts the optimal binary classification problem into a one-class classification problem23 as Display Formula

ΛAD(x)= p(x|ωB)ωBωTα.(5)
However, besides modeling the background, some other CFAR algorithms do consider a PDF model for the target-class. It is preferable to adopt such a target model, even if it merely represents a broad PDF (i.e., a weak prior assumption) rather than doing an AD24,25 as shown in Eq. (5).

In the former approach (i.e., the AD approach), the CFAR algorithm assigns PUTs to the background if it finds that the PUTs are consistent with the background distribution; otherwise, the PUTs are labeled as detected. For a desired PFA, the scaling factor α is adaptively estimated throughout the image from the boundary ring in the sliding window focused on the ROI as Display Formula

PFA=αp(x|ωB)dx.(6)
The latter approach (i.e., the one that models both the background-class and the target-class) utilizes the NP criterion as explained in Eq. (4), and the PD, for a certain target model of interest, p(x|ωT), is given by Display Formula
PD=αp(x|ωT)dx.(7)

CFAR-based methods

Be it a fixed-size or adaptive sliding window, the various CFAR methods can be viewed through two perspectives intermingled. The first is based on the method used to estimate the threshold scaling factor (for a desired PFA) in the boundary ring and/or the approach utilized for estimating the target signature (for a desired PD). There are two strategies: parametric CFAR and nonparametric CFAR. The second is based on the method used to implement the CFAR technique. There are various strategies, including CA-CFAR, SOCA-CFAR, GOCA-CFAR, and OS-CFAR. Thus, any CFAR detector can be viewed as a combination of these two perspectives: one strategy pertinent to the estimation of the threshold scaling factor and one strategy for implementing the CFAR technique. To visualize the interrelation between the various methods and strategies, refer to Fig. 3.

Under this section, we review parametric CFAR and nonparametric CFAR. Under parametric CFAR, we review one-parameter CFAR and two-parameter CFAR. Under one-parameter CFAR, we review various implementation strategies, including CA-CFAR, SOCA-CFAR, GOCA-CFAR, and OS-CFAR. Under two-parameter CFAR, we discuss the most common implementation strategy. We then briefly discuss CFAR loss. Then an interesting remark that addresses an important issue pertaining to CFAR usage is presented. Finally, the topic of nonparametric CFAR is briefly approached.

Parametric CFAR

Parametric CFAR methods can be classified into two classes: methods based only on background modeling (i.e., AD) and methods based on both background and target modeling. We briefly review these two methods here.

All the parametric CFAR algorithms that perform AD (i.e., only models the background clutter) have one thing in common. They all assume that the background clutter can be roughly modeled by a certain probability distribution, i.e., p(x|ωB). Then, to perform the CFAR detection, they estimate the model distribution parameters from the boundary ring in the CFAR stencil. This PDF model is used to estimate the threshold scaling factor α, for a desired CFAR (i.e., PFA), as the focusing window is systematically slid over the SAR image. However, variant classes of CFAR algorithms primarily differ in two aspects.

First, there is the probability distribution chosen, i.e., p(x|ωB), to model the background clutter. For example, some CFAR algorithms assume a homogeneous clutter and model the background clutter with an exponential distribution (i.e., for SAR image in the power-domain; magnitude-squared), or a Rayleigh distribution (i.e., for SAR image in the magnitude-domain). This class of distribution models is characterized by one-parameter (i.e., the mean) and thus is referred to in the literature as a one-parameter CFAR. Other CFAR algorithms model the background clutter in the SAR image with the more realistic but more complex Weibull distribution,26,27K-distribution,28 alpha-stable distribution,29,30 or beta-prime (β) distribution,31 among other models. This class of distribution models is characterized by two parameters (mean and variance, scale and shape parameters, etc.), and because of this, the CFAR algorithm is referred to in the literature as a two-parameter CFAR.

Second, there is the method used to estimate the model parameters pertaining to the detection Threshold from the boundary ring. For example, there is CA-CFAR, GOCA-CFAR, SOCA-CFAR, OS-CFAR,3234 and best linear unbiased estimator CFAR (BLUE-CFAR),26,27 among others.

The CFAR algorithms that model both the background and the target perform procedures similar to those mentioned above. However, besides estimating the background model parameters in the focusing window, they also estimate the target model parameters. Thus, the detection threshold in the sliding window is determined based on the NP criterion as shown in Eq. (4). Gan and Wang35 and Rong-Bing and Jian-Guo36 offer examples on this approach.

One-parameter CFAR

Under this section, we briefly review various one-parameter CFAR implementation strategies, including CA-CFAR, SOCA-CFAR, GOCA-CFAR, and OS-CFAR. Obviously, these same strategies are also utilized for implementing two-parameter CFAR. Accordingly, a proper understanding of these implementation strategies paves the way for better understanding two-parameter CFAR.

CA-CFAR was the first CFAR test proposed in 1968 by Fin and Johnson.3739 The adaptive Threshold is comprised of two parts. One is estimated from the boundary ring called Z, and the other is found from the corresponding PDF distribution for a desired PFA. This explains why α is referred to as the threshold scaling factor. Thus, Threshold is given by Display Formula

Threshold=αZ.(8)
The underpinning assumption is that the clutter in the in-phase (I) and quadrature (Q) channels of the radar (particularly speckle, as no interfering targets are assumed) is Gaussian. Note that the focused SAR data output from the SAR processor (i.e., level-1 processed) is a complex-valued image of the form, I+jQ. The CA-CFAR detection can be performed on the magnitude SAR image (i.e., A=I2+Q2), the power SAR image (i.e., P=A2), or the log-domain image (i.e., L=10logA2). It should be highlighted that a CFAR detector that deals with a magnitude SAR image is commonly known in the literature as an envelope (i.e., magnitude) detector or linear detector. A CFAR detector designed to handle a power SAR image is known as a square-law (i.e., power) detector. Finally, a CFAR detector designed for a log-domain SAR image is known as a log detector. Note that this terminology is also applicable to the two-parameter CFAR method. Accordingly, the clutter in the SAR image will be Rayleigh or exponential distributed depending on whether the SAR image is magnitude or power, respectively. Obviously, both the exponential distribution and Rayleigh distribution are completely characterized by one parameter (i.e., mean), thus CA-CFAR is also known as one-parameter CFAR.

Assuming a power image, and assuming the clutter is independent and identically distributed (iid), and that the PDF of a pixel xi is exponential distributed, whereby both the I and Q channels with power Z/2 in each channel (total Z), as follows: Display Formula

p(xi|ωB)=1Zexi/Z.(9)
Thus, for N reference pixels in the boundary ring (see Fig. 4), Display Formula
Γ=p(x|ωB)=1ZNi=1Nexi/Z=1ZNei=1NxiZ.(10)
From this, Z is approximated from the log maximum likelihood estimate (log-MLE) as Display Formula
ddZlnΓ=ddZ(i=1NxiZNlnZ)=0.(11)
Thus, Display Formula
μ^B=Z^=i=1NxiN.(12)
Note that Z^ is simply the maximum likelihood estimate (MLE) of the arithmetic mean of the pixels in the boundary ring in the CFAR stencil. Accordingly, from Eq. (8), Display Formula
Threshold=αZαμ^B=αi=1NxiN.(13)
The scaling factor is estimated based on Eq. (6), and for this case is given40 by Display Formula
α=N(PFA1N1).(14)
Since PFA does not depend on the average power Z in the reference cell, the algorithm is CFAR under the assumptions mentioned earlier in this section.40

In reference to Fig. 4, CA-CFAR computes the arithmetic average of the pixels in the boundary ring and then compares it with the PUT. The decision to conclude either that a PUT is detected (i.e., target-class) or not detected (i.e., background-class) is contingent upon the threshold scaling factor α, as illustrated below. The SAR image is in the (noncomplex) power domain: Display Formula

XPUTμ^BωTωBα,(15)
where μ^B is calculated based on Eq. (12) whereby N is the total number of pixels in the boundary ring (N=56 in Fig. 4), and xi is a pixel value in the boundary ring, and XPUT is the PUT.

Note that if a log-detector CFAR is used (i.e., the SAR image is assumed to be in the log-domain), one simply takes the logarithm for both sides in Eq. (15). Accordingly, the log-detector one-parameter CFAR is governed by Display Formula

XPUTlogμ^BlogωTωBαlog.(16)
The subscript log is inserted in the above inequality to differentiate it from Eq. (15) and to emphasize the fact that all the terms in Eq. (16) are calculated based on the SAR image in the log-domain.

Additionally, note that it is possible to consider more than one pixel for the PUT in the inequalities above. However, in this case, XPUT should be replaced with the ML estimate of the arithmetic average pertaining to the M PUTs given by Display Formula

X¯PUT=i=1MxiM.(17)
CA-CFAR relies on two major assumptions that pose limitations on its performance. First, the targets are assumed to be isolated by at least the stencil size, so that there is at most one target in the stencil at one time (i.e., no interfering targets are assumed). Second, the reference pixels in the boundary ring are assumed to be iid distributed and have a PDF similar to that in the PUT (i.e., homogeneous clutter). Obviously, these assumptions do not hold in many real-world scenarios. Thus, when CA-CFAR is used under circumstances different from the design assumptions, its performance suffers a significant detection loss40 (also known as CFAR loss).

SOCA-CFAR and GOCA-CFAR are variants of CA-CFAR that split the boundary ring in the sliding stencil into separate leading and lagging windows (indeed, there are four such windows, one on each side of the PUT) that are used to estimate separate statistics.41 This is depicted in Fig. 5.

Grahic Jump LocationF5 :

Four different strategies for defining the leading (a), (b); and lagging (c), (d) windows. Pixels shaded in orange illustrate a leading or lagging window in relation to the PUT.

In SOCA-CFAR, the smallest of the four mean estimates is used to perform the test, while in GOCA-CFAR, the greatest of the four mean estimates is used. Assume there are a total of N pixels in each window. Thus, there are four mean estimates: Display Formula

meantop=i=1Nxi,topN,meanleft=i=1Nxi,leftN,meanbottom=i=1Nxi,bottomN,andmeanright=i=1Nxi,rightN,(18)
where meantop, meanleft, meanbottom, and meanright are the arithmetic average estimates for the top, left, bottom, and right windows, respectively, and xi is the corresponding pixel value inside each window.

Accordingly, for SOCA-CFAR, the detection decision for power-domain (i.e., square-law) detection is Display Formula

XPUTmin{meantop,meanleft,meanbottom,meanright}ωTωBα.(19)
Similarly, for GOCA-CFAR, the power-domain detection decision is Display Formula
XPUTmax{meantop,meanleft,meanbottom,meanright}ωTωBα.(20)
Note that, similar to CA-CFAR, the threshold scaling factor α is determined based on the desired PFA and the (Gaussian) distribution used to model the clutter in the relevant reference window. Also note that the relationship between α and PFA will be different from that given in Eq. (14). For derivation of the relationship for GOCA-CFAR, the reader is referred to Chap. 7, pages 365, and 367, respectively, in 40.

SOCA-CFAR is designed to handle strong clutter returns in the boundary ring, but it is susceptible to clutter edges. On the other hand, GOCA-CFAR is capable of performing better than CA-CFAR and SOCA-CFAR at clutter edges, but its performance degrades when strong returns appear in the boundary ring. Further, compared to CA-CFAR, both SOCA and GOCA suffer from additional CFAR loss, due to the consideration of only a part of the boundary ring.

OS-CFAR was first proposed by Rohling42 to counter multiple targets in the CFAR stencil. OS-CFAR rank orders the pixels in the boundary ring according to their value. Assuming that there are N pixels in the boundary ring of Fig. 4, OS-CFAR orders them in an ascending order: Display Formula

x(1)x(2)x(N).(21)
Further, the Q’th percentile is chosen in place of the average estimate in CA-CFAR. Thus the detection decision is based on Display Formula
XPUTx(Q)ωTωBα.(22)
Similar to earlier CFAR methods, the threshold scaling factor α is estimated based on the clutter statistics in the boundary ring. In Rohling’s original work,42Q=3/4 is found to work best. However, in later work,43,44Q=4/5 is found to give better results. Obviously, the best value of Q is dependent on the type of the SAR data used. Thus, bearing these values in mind, it is desirable to empirically check the value of Q that best suits the data in use. For derivation of the threshold scaling factor α for a desired PFA under the Gaussian assumption, the reader is referred to page 372 in Richards.40

In heterogeneous/nonhomogeneous clutter backgrounds, and for contiguous targets, OS-CFAR is known to outperform CA-CFAR.38,42,45,46 However, the performance of OS-CFAR degrades during clutter transitions.45 This motivated researchers to develop this method further to handle various scenarios. Although only tested on 1-D radar data (i.e., range profiles), a variant called switched order statistics (SWOS) CFAR is designed for detecting targets in nonhomogenous clutter and/or multiple interfering target scenarios.47 This algorithm builds on a relevant method known as selection and estimation48 (SE) and is able to determine whether the cells in the boundary ring belong to homogeneous or nonhomogeneous clutter and thus adaptively adjust the detection threshold for a desired PFA. SWOS-CFAR is shown to outperform standard OS-CFAR49 and switching CFAR (S-CFAR). A generalized version of S-CFAR termed GS-CFAR is proposed, wherein it is shown that GS-CFAR yields some improvement in the detection probability in the case of interfering targets and clutter edges.50 Many additional variations of CFAR exist in 40.

Gandhi and Kassam51 show that the performance of the CA-CFAR processor approaches that of the NP detector in that the limit of the number of pixels in the boundary ring approaches infinity under the condition that homogeneity is maintained in the boundary ring. Thus, CA-CFAR achieves asymptotically optimal performance under these (theoretical) assumptions. Accordingly, CA-CFAR can be used as the baseline algorithm for comparison with other CFAR techniques.

We summarize the four basic (square-law) CFAR detectors (i.e., CA-CFAR, SOCA-CFAR, GOCA-CFAR, and OS-CFAR) in Table 1.

Table Grahic Jump Location
Table 1Summary of the four basic square-law CFAR techniques.
Two-parameter CFAR

Unlike one-parameter distribution models discussed earlier, more realistic two-parameter distribution models characterize the clutter in the boundary ring of the CFAR stencil by two parameters (mean and variance, scale and shape, etc.). Examples of two-parameter distribution models include log-normal distribution and Weibull distribution. For high-resolution SAR imagery, compound two-parameter distributions such as K-distribution, Go-distribution, and β’-distribution are typically used.

A conventional two-parameter CA-CFAR algorithm based on the log detector has the form14,52,53Display Formula

XPUTlogμ^Blogσ^BlogωTωBαlog,(23)
where αlog is the threshold scaling factor estimated for a desired PFA based on the relevant model distribution for the background clutter.

Accordingly, rearranging Eq. (23) yields Display Formula

T^hresholdlog=μ^Blog+αlogσ^Blog,(24)
where μ^Blog and σ^Blog are the ML estimates of the background mean and standard deviation, respectively, calculated from the log-domain SAR image, and given by Display Formula
μ^Blog=i=1NxiNandσ^Blog=1Ni=1N(xiμ^Blog)2,(25)
where xi is the pixel in the boundary ring, and N is the total number of reference pixels in the boundary ring.

PUT is assumed to be a single pixel. If more than one pixel is assumed, then PUT is the ML estimate of the arithmetic mean such that Display Formula

X¯PUT=i=1MxiM,(26)
where M is the number of pixels in the PUT.

However, note that this will entail replacing54σ^Blog in Eqs. (23) and (24) with σ^mlog: Display Formula

σ^mlog=σ^BlogM.(27)
An embedded underlying assumption in deriving Eqs. (23) and (24) is that the background clutter in the I and Q channels of the SAR image follows a Gaussian distribution, and thus the clutter in the magnitude-domain SAR image or the power-domain SAR image is Rayleigh or exponential distributed, respectively. This assumption entails converting the SAR image to the log-domain. In some other works, the background clutter in the SAR image is assumed to be log-normal, and thus the above equations are adopted verbatim without applying the logarithmic conversion to the SAR image.19,21,22,54

If we keep the assumption implicit in Eqs. (23) and (24) (i.e., magnitude image is Rayleigh distributed, and power intensity image is exponential distributed), the two-parameter CFAR applied to the image in the (nonlog) domain is given55 by Display Formula

XPUTμ^B1σ^BωTωBα.(28)
If more than one pixel is considered in the PUT, then a procedure similar to Eqs. (26) and (27) should be applied. Note that various CFAR combinations described in the previous section (i.e., SOCA, GOCA, OS, etc.) can also be applied to two-parameter CFAR.

CFAR loss

As it is explained thus far, the CFAR approach aims at maintaining a CFAR by locally adapting the detection threshold to the background clutter in the SAR image. However, a detection loss, commonly referred to as CFAR loss, is the price paid for this threshold adaptation. This is due to the fact that, in real-world applications, the noise level in the boundary ring is not constant and/or the number of reference pixels used in the estimation is not large enough. Further, CFAR loss can be viewed as the required increase in the signal to noise ratio (SNR) in order to maintain the desired PD.56 The value of this CFAR loss is dependent upon a number of factors, including CFAR method used (e.g., CA, GOCA, SOCA, etc.), number of pixels in the boundary ring, PFA, target statistics, clutter statistics, and noise. Antipov and Baldwinson57 offer a notable work on this topic, though applied to 1-D radar data.

A universal curve for CFAR loss in a single-hit detection for steady or Rayleigh target56 is depicted in Fig. 6. The parameter χ is given by Display Formula

χ=logPFA.(29)
The CFAR ratio is given by Display Formula
CFARRatio=χmeff,(30)
where meff is the effective number of reference pixels in the boundary ring, given by Display Formula
meff=m+k1+k.(31)
Values of k for different CFAR detectors are provided in Table 2.

Grahic Jump LocationF6 :

Universal curve of CFAR loss in single-hit detection for steady or Rayleigh target (See Refs. 56 and 57).

Table Grahic Jump Location
Table 2Values of k under different CFAR detectors (56).

We now present a final word on the log-detector CFAR. Indeed, caution should be exercised when dealing with log-CFAR. As reported in 40 log-CFAR introduces an additional CFAR loss into the process. To circumvent this CFAR loss, the size of the CFAR stencil, more precisely the boundary ring in the stencil, needs to be increased by 65 percent58 by following Display Formula

Nlog=1.65N0.65,(32)
where N is the number of pixels used for the nonlog CA-CFAR detector, and Nlog is the number of corresponding pixels required for the log CA-CFAR detector to circumvent the CFAR loss due to using the log-detector.

Indeed, despite this additional hurdle, log-detection CFAR is shown to be preferred over linear threshold CFAR processing for typical nonhomogeneous clutter conditions where background clutter surrounding the target is contaminated by other targets, bright clutter pixels, etc.59,60

Final remarks on parametric CFAR

This section highlights the fact that the CFAR detectors described earlier that are originally designed under the Gaussian assumption (i.e., the background clutter distribution in the I and Q channels) are typically applied in the literature to nonGaussian clutter. Typically, a suitable clutter model (Weibull distribution, K-distribution, β’-distribution, etc.) is adopted, wherein the distribution model parameters are estimated from the boundary ring in the sliding stencil and are used to estimate the scaling factor (α) for the desired PFA.

In 19, the two-parameter CFAR scheme is originally designed under Rayleigh/Gaussian assumptions (similar to the one presented earlier) and applied to nonGaussian data in high-resolution SAR images. In Armstrong and Griffiths,61 the performance of two-parameter CA-CFAR, GOCA-CFAR, and OS-CFAR (originally designed under Gaussian assumption) is evaluated under K-distributed clutter. In Refs. 31 and 62 two-parameter CA-CFAR (originally designed based on Gaussian assumption) is applied to β’-distributed high-resolution SAR data. Additional examples are provided in Refs. 53 and 54. Obviously, such CFAR detection schemes are applied to clutter distributions that are different from those on which the CFAR detector was originally designed. Subsequently, the conclusion is reached that CFAR techniques prove to be effective regardless of this fact, which explains their popularity.63 However, CFAR loss is the price paid for this process.40

Nonparametric CFAR

In the parametric CFAR methods explained in the preceding sections, the background clutter and/or the target pixels are assumed to obey a certain distribution, and the pixels in the reference window (i.e., boundary ring) are used to estimate the corresponding model parameters. Nonparametric CFAR algorithms do not assume any prior model for the background or the target. Rather, they use nonparametric methods18,64 to directly infer the model from the SAR data. An example on a nonparametric CFAR using kernel density estimation (KDE) for inferring the background and the target models is proposed in Gao.65

Nonrectangle-shaped-window methods

While most detection techniques reported in the literature rely on a rectangle-shaped hollow stencil with a suitable size and a guard ring, there are some other methods that replace the rectangle-shaped window with another shape. One such example is provided in 66, where the rectangle-shaped CFAR window is replaced with a 2-D gamma kernel. The method utilizes a CFAR detection strategy and is referred to as gamma-CFAR. Similar to traditional rectangle-shaped-window CFAR, the CFAR threshold in gamma-CFAR is estimated from the radial pixel intensity information around a PUT. Further, gamma-CFAR has a free parameter which can be used to estimate the size of its region of support and its standard deviation required for the CFAR test. Although it is not implemented in 66 it is stated that this free parameter can be set adaptively. The gamma stencil is also applied in a non-CFAR fashion based on a quadratic Gamma discriminant (QGD) that has eight free parameters.66 Test results show that QGD outperforms gamma-CFAR. Extension of QGD to an artificial neural network (ANN) based on a multilayer perceptron (MLP) is provided in 67.

Non-CFAR methods

Besides CFAR-based methods, there are other approaches that do not use CFAR. For example, in Ouchi et al.,68 the detection is based on a coherence image produced from the multilook SAR image via cross correlation between two SAR images extracted by moving windows of a small size over the original image. The method can detect objects buried in a speckle. In Howard, Roberts, and Brankin,69 the detection is based on genetic programming. Relevant work is provided in Conte, Lops, and Ricci,70 although applied to 1-D radar data, where the detection is based on a generalized likelihood ratio test (GLRT) in a K-distributed clutter. The null hypothesis (Ho) represents the K-distributed clutter, and the alternative hypothesis (H1) is modeled as being a compound of target signal and a K-distributed noise.

Multifeature-Based Taxon

All the aforementioned methods for target detection are single-feature-based in that they distinguish the target pixels from the background only on the basis of the level of pixel brightness (i.e., RCS). This poses a limitation on these methods, more significantly, in ROIs with heterogeneous clutter and/or contiguous targets.

Methods under the multifeature-based taxon try to circumvent this drawback by basing the detection decision on a fusion of two or more features. Obviously, this taxon can utilize a suitable method among those presented under the single-feature-based taxon and incorporate additional features besides RCS. Methods that fall under the multifeature-based taxon can be broadly classified into two major sub-taxa: those that utilize arbitrary user-chosen features and those that rely on systematic multiresolution analysis. Examples of arbitrary user-chosen features are provided in Refs. 9 and 71, wherein decisions on ROIs in the SAR image are based on a fusion of three multistage features extracted in parallel from the SAR image, namely CFAR features, variance features, and extended fractal (EF) features. Obviously, this approach is not purely CFAR. Another relevant example of the multifeature method is provided in 72. Further, in Subotic et al.,73 parametric circular complex Gaussian models are utilized for both target and clutter. However, unlike the traditional CFAR approach, which works on a single resolution, the resolution of the SAR image in this approach is varied to produce multiple features. It is based on the conjecture that target signatures, when viewed as a function of resolution, are sufficiently different from clutter. This allows detection performance gains over single-resolution detection algorithms.

This perspective motivates the more systematic multiresolution analysis methods, which can be broadly classified into space-scale-based and space-frequency-based. Space-scale methods produce space-scale features based on the wavelet transform, including the DWT, and CWT. In 74 the detection is based on DWT that yields a spatial correlation of the sub-bands that attenuates the background noise and enhances the structured pattern of the target signature. A description of the more relevant detection strategies based on CWT can be found in Antoine et al.75

Prime examples of methods that utilize space-frequency features include linear space-frequency methods, such as the S-transform,76 and bilinear (also known as quadratic) space-frequency methods, such as Cohen’s class distributions77,78 (Wigner distribution, Wigner-Ville distribution, pseudo-Wigner-Ville distribution, etc.).

Expert-System-Oriented Taxon

Expert-system-oriented target detection is a multistage (two or more stages) AI approach that bases the detection process on exploitation of prior knowledge about the imaged scene and/or target(s). Prior knowledge is captured via context utilization. In the broader sense, context here refers to all the available means that can help to convey information about the background clutter and/or target(s) of interest in the scene. Such means include image segmentation, scene maps, digital elevation model (DEM), previously gathered data, and geographical information system (GIS).

In its simplest form, context utilization in CFAR can be realized via unsupervised/semisupervised SAR image segmentation. Typically, prior to applying CFAR to the SAR image, the SAR image is segmented to extract an image structure map. Then, the conventional CFAR method of choice is aided with this map to enable it to adaptively select the suitable area over which the background statistics can be properly estimated. Further, smaller segments can be labeled as potential targets, while larger ones can be labeled as background. An example for utilizing annealed segmentation with one-parameter CFAR for SAR target detection is provided in McConnell and Oliver.79

A relevant work is termed variability index CFAR (VI-CFAR), although applied on 1-D range data.80,81 Based on the estimations of the mean in the boundary ring of the stencil, VI-CFAR switches between one of the following three CFAR methods: CA-CFAR, SOCA-CFAR, and GOCA-CFAR. This approach tries to benefit from the strength of each CFAR method through deployment of the switching scheme.

Another relevant algorithm is reported in Gao et al.62 The algorithm builds on the beta-prime CFAR (β’-CFAR) algorithm31 reported earlier in this paper. A binary index map is created based on globally thresholding the input SAR image. The index map is comprised of zeros assigned to pixels in the input SAR image that are found to be less than a predetermined global threshold, and ones assigned to pixels found to be greater than the global threshold. Then, a sliding window stencil is placed over the image (i.e., pixel by pixel) where the parameters of the Go-distribution (i.e., Go-distribution reduces to β’-distribution for a single look SAR image; more on this is given in Sec. 5.2) are estimated from the pixel values in the boundary ring correspond to 0 in the index map. The size of the stencil is chosen based on the prior knowledge of the target size as described in Salazar.31 Then, a local threshold in the stencil is determined based on the parameters estimated, and a decision on the PUT is taken. The window is then systematically slid to subsequent pixels in the image until the whole image is scanned. Further refinements on detections are achieved by placing a constraint on the size and allowable distance between detections.

More robust expert-system-oriented approaches utilize a mixture of multiple different CFAR experts (CA-CFAR, OS-CFAR, etc.), each of which is specialized to handle a suitable kind of clutter. The expert-system-oriented system uses available context information, extracted by one or more means (as explained earlier in this section), to assign the CFAR experts to suitable regions in the SAR image. Unfortunately, there is a lack of work published on this approach for SAR imagery. However, there are works published on 1-D radar data. One such example8284 was implemented by the U.S. Air Force Research Laboratory.

Rimbert and Bell43,44 present another interesting work. It is motivated by the observation that, in homogeneous regions (i.e., locally stationary clutter), a larger reference window provides a clutter estimate that has a mean with smaller variance than a similar estimate based on a smaller reference window. Conversely, a smaller reference window provides a more reliable mean estimate in nonhomogeneous regions. Accordingly, a target detection scheme that adapts the reference window size and selects one of two CFAR detectors (i.e., CA-CFAR, and OS-CFAR) based on the type of the reference region is proposed. The proposed detection scheme checks the homogeneity of the reference region using a goodness-of-fit statistical test with an assumed parametric model for the clutter. One can think of this process as producing a structure map for the reference window (i.e., not for the whole image, as discussed in this section for the segmentation-based method). A CFAR method termed cell under test inclusive CFAR (CI-CFAR) is also introduced, which operates in a manner similar to OS-CFAR but combines the PUT with the clutter pixels in the reference window. Although the authors reported in one work43 that CI-CFAR outperforms OS-CFAR, in another work,44 the authors acknowledge that this conclusion is incorrect, due to the use of a simulation with errant detection threshold parameters. It should be noted that the results presented are not based on 2-D SAR data. Further, the analysis was based on the assumption that both the clutter and the target obey the central limit theorem (CLT) (i.e., Gaussian), which is a nonrealistic assumption for high-resolution SAR imagery. Additionally, reference stencils used are not hollow, and they do not consider any guard regions.

Finally, any relevant detection method that utilizes any form of intelligence/inference is of particular interest and to fit under this taxon.

A comparison between selected examples, most of which are from amongst those cited in Sec. 3, on the detection module pertaining to SAR imagery is provided in Table 3. The choice of the examples was carefully made to cover the different methods under each taxon. Comparison aspects include SAR image type, feature(s), clutter/target type, clutter model (if applicable), and target model (if applicable). The comparison does not attempt to assess the algorithm performance based on reported PD and PFA, as this is infeasible, given the variant types of methods and data used, as well as the different sensor characteristics and operating conditions. Rather, the aim of this comparison is to show the major differences between the various methods and depict their applicability to certain scenarios.

Table Grahic Jump Location
Table 3Comparison between selected detection modules [T=Taxon, ST=Sub-taxon].

Obviously, target detection based on parametric modeling of SAR imagery is the most popular in the literature. This discussion focuses on parametric CFAR methods that utilize stochastic models for modeling the background clutter. Primarily, the issue of the suitability of these models to represent the SAR data is briefly discussed. This is followed by a concise summary of the popular multiplicative (also known as compound) SAR data models, conditions of their applicability, and the interrelation between them. Finally, our focus is shifted to parametric CFAR. We tame CFAR and present some novel discussions from two different perspectives: the signal processing perspective and the statistical pattern recognition perspective.

On the Suitability of SAR Data Models

In Table 3, there are various parametric clutter models used, including log-normal-distribution, K-distribution, and exponential distribution. Accordingly, in Table 4, we briefly summarize some of the major statistical distributions, along with the backscatter types in SAR images they are typically used to model.

Table Grahic Jump Location
Table 4Major statistical distributions and suitable modeling phenomenon.

As is evident in Table 3 (and relevant works published in the literature), in many cases, similar distributions are being randomly chosen to model the clutter at various sensor characteristics, such as frequency, polarization, imaging mode (e.g., Spotlight, Stripmap, ScanSAR, etc.), and resolution. Obviously, the choice of a proper model for the clutter backscatter depends, not only on the clutter type, but also on these sensor characteristics. Several works89,95,96 have noted that the suitability of a certain distribution to model a certain clutter depends on the data being used, as well as the corresponding sensor characteristics and operating conditions. Surprisingly, this issue is generally overlooked in many works published in the literature. In many such works, the justifications for choosing some model (e.g., K-distribution) is merely based on the assumption that the distribution model is found suitable for a certain clutter type (e.g., ocean clutter) in some published work, and thus it can be automatically adopted for a similar clutter. Indeed, such a conclusion can be misleading.

Accordingly, prior to randomly opting for a popular parametric distribution to model the clutter, one should consider validating the applicability of the distribution on the data using a suitable goodness-of-fit technique.97,98 One such validation approach for SAR data is based on the Cramer-Von Mises (CVM) distance97 as presented in di Bisceglie and Galdi.26 The normally used Kolmogorov-Smirnov test is discarded, because the independency assumptions of its usage are violated, due to the dependency of the SAR data being non-Gaussian distributed and generally correlated.97 Thus, the CVM method is used to measure the distributional distance between the design cumulative distribution function (CDF) model and the empirical CDF estimated from the available high-resolution SAR image. The design distribution that scores a minimum distance within some threshold is typically chosen. This procedure is demonstrated in di Bisceglie and Galdi26 on Spotlight (9.6 GHz, HH polarized, geometric resolution of 0.4m×0.4m) SAR data (with a specific clutter type) pertaining to the Rayleigh and Weibull distributions. On the SAR image used in di Bisceglie and Galdi,26 the Weibull distribution is found to be a more suitable model for the clutter.

One final observation on this matter is that, though the CVM test was used in the work mentioned above, CVM is not the best technique when it comes to characterizing tailed SAR data. Indeed, the k-sample Anderson-Darling test99 offers a better procedure, as it places more weight on observations in the tail of the CDF distribution. Both the CVM test and Anderson-Darling test belong to the quadratic class of empirical distribution functions (EDF) for statistical tests.99 Finally, information-theoretic approaches for characterizing the goodness-of-fit can be found in 100

Understanding the Multiplicative SAR Data Models

In any target detection scheme that depends on parametric modeling (e.g., the popular CFAR-based), the selection of an appropriate probability distribution to model the pixels in the SAR image (i.e., radar backscatter) is a must, because the thresholding operation in any such detector is dependent on the clutter distribution. In cases where the random scatterers in a resolution cell in the SAR image have sizes on the order of the wavelength of the radar signal, the total backscatter can be modeled as the sum of isolated returns in the cell.101 This invokes the CLT, wherein the I and Q components of the total complex-valued backscatter can naturally be assumed to be normally distributed. This implies that the total backscattered amplitude and phase can be modeled as Rayleigh and uniform distributions, respectively. Thus, the power in each resolution cell is modeled as an exponential distribution. Conversely, in high-resolution SAR, the above mentioned assumptions are violated, because the number of random scatters in a resolution cell is not large, and thus the CLT cannot apply. This renders the clutter non-normally distributed, which motivates the need for a suitable model.

The multiplicative model (also known as the compound model) for SAR image formation has been popularly used in the literature to model the clutter background. The model is based on the hypothesis that the SAR image is formed from the product of a backscatter and speckle random processes as Display Formula

Z=X×Y,(33)
where X and Y are two independent random variables that represent the backscatter and speckle, respectively. X is often assumed to be a positive real variable, whereas Y is either complex or positive real, depending on whether the image is in the complex or intensity/magnitude domains. The product in Z models the observed SAR image.

Typically, for a single look (i.e., n=1) SAR image, the complex speckle Y is characterized as bivariate normal density for complex imagery, which reduces to the exponential distribution in the power or intensity domain. Further, for multilook imagery (i.e., n>1), the two-parameter gamma distribution Γ(α,λ) characterizes the speckle in the power domain, and this reduces to the square root gamma distribution Γ(α,λ) in the magnitude domain. Depending on the type of the background clutter (i.e., homogeneous, heterogeneous, or extremely heterogeneous), and the pertinent sensor characteristics (i.e., operating conditions) such as frequency, polarization, and gazing angles, several different distributions are used in the literature to model the backscatter X. For each case, the manner in which X (and subsequently Z) manifests itself depends31,94 on whether the SAR image is single-look (i.e., n=1) or multilook (i.e., n>1).

First, for homogeneous regions and a single look SAR image, X is typically modeled as a constant that equals the average power in the homogeneous region (i.e., C=2σG2). Accordingly, the power-domain SAR image Z is modeled as exponential distributed: Display Formula

Zexp(αλ).(34)
Similarly, for a multilook SAR image (i.e., n>1), α=n, and λ=n/(2σG2). However, the power domain SAR image Z becomes gamma distributed:Display Formula
ZΓ(n,nλα).(35)
Second, for heterogeneous regions, the backscatter X is not modeled as a constant. Rather, it is modeled as a gamma distribution Γ(α,λ), or a square root gamma distribution Γ1/2(α,λ), for power-domain and amplitude-domain SAR imagery, respectively. This yields the K-distribution model, K(α,λ,n) for any number of looks n.

Third, for extremely heterogeneous regions, the G-distribution93 is typically used. Unlike the K-distribution, the G-distribution uses the square root of the generalized inverse Gaussian distribution to model the backscatter X for both homogeneous and heterogeneous backgrounds in the magnitude-domain: Display Formula

XN1(α,γ,λ).(36)
The speckle model is left unchanged as provided earlier. This model is the most generic, and the previous models are special cases of it. Indeed, N1(α,γ,λ) leads to the following three special cases. First, the square root of the gamma distribution leads to the K-distribution. Second, the reciprocal of the square root of the gamma distribution leads to the Go-distribution. Third, a constant leads to a scaled speckle (i.e., the homogeneous case), as explained in Eqs. (34) and (35), for single-look and multilook SAR imagery, respectively.

With the same number of free parameters (i.e., two parameters) as the K-distribution, the Go-distribution can model extremely heterogeneous regions that the K-distribution cannot model. Finally, the Go-distribution reduces to the beta-prime distribution, β(α,γ)