2.21.4.1.1.1 SAR image spectral content

As depicted in Figure 21.46, a SAR measurement consists in repeatedly emitting a signal, , in the across track direction and receiving the echo from the observed scene, , at different locations along the acquisition track. A scatterer located at coordinates is observed for different values of the azimuth look angle, , defined by , defined being the varying radar-scatterer distance. The range of the azimuth angle is defined by the antenna aperture, whereas the emitted signal is characterized by a bandpass spectrum centered around a carrier frequency , with bandwidth .

After base-band conversion, the received signal can be focused to produce a 2-D coherent image in the azimuth-range domain, , that represents a reconstruction of the coherent reflectivity of the scene. Under simplifying assumptions and considering ideal acquisition conditions [21], a coherent SAR image may be formulated as convolution of the scene coherent reflectivity, and the SAR 2-D impulse response, and may be represented in both spatial and spectral domains as:

(21.139)

where is the 2-D Fourier transform of the SAR image and represents the convolution operator. The spectral coordinates , representing two-way wavenumbers in range and azimuth, are illustrated in Figure 21.47 and can be formulated as [43]:

(21.140)

where is the emitted signal wavenumber and can be related to the electrical frequency using, , through the wave propagation velocity, , as .

For an ideal scene, whose reflectivity is uniformly distributed over the spectral domain, the resolutions of the SAR image are driven by the impulse response of the SAR system and are given by

(21.141)

where stands for the processed azimuthal aperture, whose maximal value is set by the acquisition antenna characteristics.

It is worth noting that the direct relation between a SAR image and the reflectivity of scene in (21.139), as well as the physical meaning of the spectral coordinates in (21.140) are valid when dealing with coherent Single Look Complex (SLC) SAR data sets, i.e., each pixel of the SAR image corresponds to a complex number whose modulus is proportional to the focused reflectivity and whose absolute phase depends on the observed medium as well as on the measurement phase history. Transforming an SLC image to an incoherent one, like an intensity image , involves an irremediable loss of information and interpretation.

2.21.4.1.1.2 Time frequency decomposition

The T-F decompositions technique selected here is based on the use of a 2-D windowed Fourier transform, or 2-D Gabor transform. This kind of transformation permits to decompose a two-dimensional signal, , with a 2-D location, into different spectral components, using a convolution with an analyzing function , as follows [44]:

(21.142)

where indicates a position in frequency, and represents the decomposition result around the spatial and frequency locations and . The application of a Fourier transform to (21.142) shows that the spectrum of is given by the product of the original signal spectrum and the transform of the analyzing function shifted around the frequency vector :

(21.143)

It is clear from Eqs. (21.142) and (21.143) that this time frequency approach may be used to characterize, in the spatial domain, behaviors corresponding to particular spectral components of the signal under analysis, selected by the analyzing function . Among the wide variety of existing TF analysis methods, the simple atomic decomposition selected in this study presents some interesting properties. It is linear, and hence preserves the coherence and energy of signals, it is not affected by artifacts related to cross-terms and may be inverted, i.e., depending on the analyzing function may be reconstructed from a set of TF samples , provided that some sampling conditions in spatial and spectral domains are satisfied. The resolutions of the analysis in space and frequency are not independent, and their product is fixed by the Heisenberg-Gabor uncertainty relation, given in 1-D by [44]

(21.144)

This relation specifies that the space-frequency resolution product equals a constant , determined by . An analyzing function with an excessively narrow spectral bandwidth would involve an excellent spectral resolution but might then lead to a meaningless analysis in the space domain owing to a bad localization, i.e., increasing the description ability of the analysis in one domain worsens its accuracy in the dual one. The nature of the analyzing function is generally chosen so as to preserve resolution while maintaining sufficiently low side-lobe amplitudes in the space domain.

2.21.4.1.1.3 Time-Frequency decomposition of SAR images

The timefrequency approach presented here deals with processed SAR images, denoted with , rather than raw data. This type of data is better accessible to common users and is generally processed through compensation procedures to reduce the effects of acquisition errors. In practice the simple SAR image model given in (21.139) needs to be completed in order to account for additional weighting terms, mainly due to the antenna pattern and side-lobe reduction functions, as [45]

(21.145)

with . The synopsis of the TF decomposition based on the spectral definition on (21.143) is given in Figure 21.48.

The first step consists of correcting for potential spectral imbalances, represented by in Eq. (21.145), in the original, full-resolution SAR image. This can be achieved by calculating average image spectra in range and azimuth and then multiplying the full-resolution spectrum by the inverse of the estimated 2-D weighting function. The TF decomposition is then conducted by multiplying the corrected spectrum by the Fourier transform of the analyzing function and going back to the spatial domain. The resulting still focused SAR image has a lower resolution than the original SAR data and depicts the scene behavior over the 2D frequency domain located in the neighborhood of . As it might be observed in (21.145), the use of processed data limits the spectral exploration range to the one of the reference function used for raw data processing and focusing. In order to emphasize the physical interpretation of coherent SAR image analysis, one may simplify the wavenumber expressions given in (21.140) using narrow beamwidth and bandwidth approximations

(21.146)

Time-Frequency decomposition in the azimuth direction consists of deriving a set of images containing different parts of the SAR Doppler spectrum with a reduced resolution, but corresponding to different azimuth look angles. This kind of analysis can be applied to detect objects or media with anisotropic behaviors, like scatterers with complex geometrical structures, human- made objects, or natural media having periodic structures in the case of agricultural areas, or linear alignments of strong scatterers [45]. In the range direction, TF analysis permits to compare the response of a scene observed at different frequencies, contained within the emitted signal spectral domain, and can be used to detect and characterize media with frequency-sensitive responses, like resonating spherical or cylindrical objects, periodic structures, or coupled scatterers with interfering characteristics.

Polarimetric SLC SAR images can be easily decomposed by applying the presented approach independently over each polarization channel. Usual polarimetric representations may then be reconstructed to study the polarimetric behavior of a scene around specific spectral locations.

(21.147)

where spatial locations, , have been omitted.

2.21.4.1.2.1 Discrimination of non-stationary POLSAR responses

Discrete Time-Frequency decomposition in range and azimuth. As shown in (21.147), the timefrequency decomposition can be applied around any frequency location inscribed within the rangeazimuth frequency range defined by the SAR transfer function . Nevertheless, it is often useful to first process the analysis around a limited (discrete) set of frequency locations to appreciate the global behavior of the scene under observation and emphasize changes from one subspectral image to another by minimizing their correlation, The polarimetric SAR data set under study has been acquired by the DLR E-SAR sensor, at L band, over the Alling test site in Germany. The original image resolution is 2 m in range and 1 m in azimuth, corresponding to an azimuthal variation of the look angle of approximately and a chirp bandwidth of 75 MHz. Figure 21.49 shows the full- resolution span image corresponding to the total polarimetric backscattered power. The considered scene is mainly composed of agricultural fields and forest. An urban area is located at the bottom left corner of the image.

The decomposition in the azimuth direction is performed using independent subspectra and keeping the range resolution to its original value. Figure 21.50 shows results obtained over an area corresponding to plowed fields. Images of the span, H, and parameters are represented for different azimuthal look angles and for the full-resolution case. It can be observed in Figure 21.50 that large variations in the scattering mechanism nature, , and degree of randomness, H, occur while the azimuth look angle changes. For particular azimuth look angles, some fields show a sudden change of behavior: the span reaches a maximum value, whereas the polarimetric indicators H and are characterized by low values. The stripes in the span image, indicating that coherent constructive and destructive interferences occur within the pixels, are characteristic for Bragg resonant scattering over periodic surfaces [45].

Other types of media may also have nonstationary polarimetric features during the azimuthal integration. It was observed that some point targets and linear structures, such as diffracting edges or road berms, have significant backscattering pattern variations as the look angle changes. In particular, metallic link fencing was found to present a scattering mechanism ranging from single-bounce up to double-bounce scattering, depending on the SAR azimuthal look angle. In general, nonstationary targets have strongly anisotropic shapes, or facets acting like directional scatterers, involving changes in the underlying scattering mechanism and in the total backscattered power.

In contrast, forested areas have a stationary behavior during the SAR integration. Backscattering from forested areas at L band is known to be dominated by volume diffusion, which corresponds to the scattering over randomly distributed anisotropic constituents. The coherent integration of the randomly scattered waves leads to a response that is characterized by a high intensity and low degree of polarization, but with isotropic behavior.

Detection of non-stationary polarimetric TF behaviors. Each pixel of the SAR scene is associated with a set of R independent target vectors, with , derived from independent range-azimuth subspectra, i.e., subspectra selected using non-overlapping functions . Under the classical speckle affected scattering hypothesis, these target vectors follow independent complex Gaussian multivariate distributions, . The stationary aspect of the scattering behavior of each pixel may then be studied by comparing the second order statistics of for different spectral locations, i.e., by testing the following hypothesis:

(21.148)

As it is shown in [46] this hypothesis can be tested easily using sample coherency matrices obtained from independent realizations or looks of each TF target vector, :

(21.149)

The TF stationary behavior of is evaluated by means of a Maximum Likelihood (ML) ratio, , built from the R independent sample coherency matrices as follows:

(21.150)

Replacing the likelihoods in (21.150) by their expression and the expectations by their ML estimates, on gets the following simple expression [45]

(21.151)

The hypothesis is accepted and the pixel under test is considered to have a stationary polarimetric TF behavior, with an arbitrarily chosen probability of false alarm , if , where the relation between the threshold value and the probability of false alarm, , has been derived in [45]. The ML ratio and nonstationary pixel map shown in Figure 21.51 indicate that an important number of pixels have a nonstationary behavior during the duration of the SAR acquisition. Most of the varying scatterers belong to agricultural fields affected by Bragg resonance. Complex targets and diffracting edges, whose scattering characteristics highly depend on the observation position, are discriminated over built-up areas and linear alignment of scatterers.

The ML ratio based detection approach may be further developed to determine nonstationary scattering behavior position in the range Doppler spectrum by comparing the contributions of each subspectrum image in the global ML ratio information [45]. It can be observed from the localization results displayed in Figure 21.52 on many fields affected by Bragg resonance that some groups of pixels, belonging to the same field, have a maximum anisotropic behavior in different subapertures. This is a consequence of the sliding effects of Bragg resonance on periodic structures, that is described in the next section.

2.21.4.1.2.2 Analysis of Bragg resonant scattering over natural soils using a polarimetric 2-D TF continuous decomposition

In the scene under examination, Bragg resonance over agricultural fields is an important source of polarimetric variations. Bragg resonance is due to the coherent summation of simultaneously constructive contributions from a set of scatterers and is likely to happen during the observation of periodic surfaces or randomly irregular surfaces with a strong periodic component, as described in 1-D in Figure 21.53 A random surface, , with a quasi-periodic component in the direction, can be described as

(21.152)

where and are the spatial period and amplitude, respectively, of the periodic component of , and the random perturbation term, , corresponds to an isotropic stationary random rough surface. This component is fully described by , the standard deviation of its zero mean Gaussian height probability density function, and , its correlation function. The Bragg resonance condition can be written as a function of the incident wavelength, , as

(21.153)

where corresponds to the local amplitude of the ground wave vector at the, is an unknown integer number indicating the mode of the resonance, is the local angle of incidence and the azimuthal angular difference between the observation position and the normal to the rows of the periodic surface. In the case of SAR measurements, can be decomposed as , where represents the orientation of the surface with respect to the normal to the SAR platform flight track, and with the angle of observation in the azimuthal spectrum, as defined in (21.140). Yueh et al. [47] developed several approaches to model the scattering of electromagnetic waves from randomly perturbed periodic surfaces. Their study reports that the influence of the resonating modes on the total backscattering response varies significantly with the surface parameters. As it can be seen in Figure 21.53, for a large surface correlation length, , and for low values of the azimuth orientation angle, , almost all the intensity peaks corresponding to different resonance modes can be discriminated. As increases, the scattering pattern becomes smoother and only a few dominant resonance peaks can be observed. In the presence of resonance, the co-polarization returns and have almost identical values, characterized by a high intensity. As the resonant effect decreases, i.e., for high values of , these polarimetric channels have distinct responses, with a significantly reduced amplitude. According to the resonance condition enounced in (21.153) similar anisotropic fields with different locations in range, , or differently oriented, i.e., with different values may resonate at different azimuthal frequencies. If the resonance conditions cannot be satisfied for any azimuthal angle within the antenna aperture or if the surface scattering characteristics do not show a resonance peak, they also might not resonate at all [48].

Moreover, some fields may have parts resonating at different positions in the azimuthal frequency domain due to the joint dependence of the resonance condition on the incidence and azimuth angles, as seen in (21.153). This phenomenon is illustrated in Figure 21.54, where the location of a resonance peak is plotted as a function of the range and azimuth frequencies. As the azimuthal look angle, , varies, the set of incidence angles satisfying Eq. (21.153) changes, leading to the apparition of sliding resonating stripes in the . The width of the resonating stripes is fixed by the width of the analyzing function in the azimuth and range directions, or equivalently .

A range-azimuth continuous Time-Frequency analysis is performed over three points , located at different range positions inside a plowed field [48]. As depicted in Figure 21.55, results can be represented, for each point, in the rangeazimuth frequency plane. The results of the timefrequency analysis, as shown in Figure 21.55, demonstrate that all three points under investigation do not have a stationary range and azimuth scattering behavior. Some couples show high span values corresponding to low and . These observations agree with the predictions of the scattering model developed by Yueh et al. [47]. As the surface resonates, the co-polarization signals tend to be similar, involving a low value, typical for surface reflection. This scattering mechanism is weighted by a strong intensity and dominates secondary intensities, potentially corresponding to multiple scattering terms, and results in a very low entropy value. This nonstationary behavior was found to have a preponderant influence on the polarimetric properties of resonating field at full resolution. Here, and values are significantly lower than those for similar fields that remained unaffected by Bragg resonance. The oblique resonating stripes, shown in the different range frequency planes in Figure 21.55, illustrate well the dependence of the resonance condition on both range and azimuth frequencies, as shown in Figure 21.54.

It can also be observed that as the incidence angle increases, from to , the oblique resonating stripe slides from low azimuth frequencies to higher ones. This displacement of the resonance locations is due to the dependence of the Bragg condition on the incidence angle and corroborates the analysis of the Bragg resonance as presented in Figure 21.54. Polarimetric indicators of pixels that do not belong to resonating stripes are unaffected by the Bragg resonance and have values similar to those observed over stationary fields.

2.21.4.1.3.1 Detection of point-like scatterers using the Internal Hermitian Product (IHP)

This approach, developed by Souyris et al. [49], was the first of a series or works on coherent TF analysis that fully exploit the physical nature of SAR signal together with polarimetric diversity. Its objective is to assess the joint use of the magnitude and the phase of a SAR polarimetric image for point target detection and analysis. The detection principle is based on the fact that scattering by point like scatterers is a coherent process, i.e., during the SAR integration waves follow a specular path. Their T-F response, after an adequate phase compensation should ideally be constant, in practice remain coherent. Over speckle affected environments, scattering is due to wave diffusion by a large number of scatterers in a random and spatially uncorrelated way. In consequence their spectral response should show a low level of correlation.

Single polarization IHP. A natural way to detect the presence of a correlated signal, i.e., a target, embedded in uncorrelated noise, i.e., the clutter, is to compute the normalized correlation between two samples:

(21.154)

In an ideal configuration, , where represents the ideally constant target response and the uncorrelated clutter contribution if the samples originate from independent spectra. In this case the TF normalized correlation writes

(21.155)

where it was assumed that the variance of the clutter response was constant over the spectral domain. The correlation is than a function of the Signal to Clutter power Ratio (SCR), i.e., when a coherent object with a strong responses is illuminated, it can be detected by thresholding . As reported in [49], such an approach may lead to poor result if the SCR is not high enough and another approach is proposed, based on the IHP, defined as

(21.156)

It is simply the un-normalized correlation between two signal samples. As it is shown on Figure 21.56 this approach permits to improve the contrast between a target and its background. One may argue that such an approach may be more associated to strong scatterer selection, based on their TF coherence properties, rather than detection, since the amplitude IHP is not normalized.

Polarimetric IHP. The IHP concept is exported to the polarimetric case [49] using the definition of the polarimetric correlation used in polarimetric SAR interferometry (POL-inSAR) [15]

(21.157)

where , and is a projection unitary vector that permits to select one or a linear combination of polarization channels. This vector is then tuned over the space of unitary 3-element complex vectors to find the optimal value, , that maximizes . The polarimetric properties of the target response are then given by that may be processed through classical decomposition techniques to obtain specific polarimetric indicators. An example of application to polarimetric SAR data is given in Figure 21.57.

2.21.4.1.3.2 Detection of coherent scatterers and their polarimetric characteristics

Persistent Scatterers (PS), whose response remain constant along time, are of particular interest for differential interferometry applications to subsidence monitoring [50]. Since it is likely that scatterers having stable TF responses, i.e., when observed from different positions and different frequencies, remain stable in time too, studies were led to investigate which types of constructions or objects actually behave as Coherent Scatterers.

2.21.4.1.3.3 CS detection based on TF entropy

In [51] the normalized correlation approach of (21.154) is compared to a new stability indicator, the TF entropy, . Unlike (21.154), this indicator can handle more than two samples at a time. The signal of a given polarization channel is sampled at different spectral locations, over independent or slightly overlapping spectral domains to create a TF signal vector, and estimate its covariance matrix, :

(21.158)

Similarly to the single-image polarimetric case, a TF entropy can be computed as [51]

(21.159)

where is the eigenvalue of . If different TF samples are maximally correlated, the TF entropy equals , whereas for totally uncorrelated signals having the same amplitude, tends toward . The analysis conducted over the city of Dresden indicates that many CS can be found over all the polarimetric channels (Figures 21.58 and 21.59). In the central part of the image is located the city center of Dresden, whereas in the central upper part a park with forested and vegetated areas can be recognized. Note that individual buildings and building blocks oriented parallel to the azimuth direction are characterized by a strong dihedral component, while buildings/blocks that have an orientation angle with respect to azimuth flight direction are characterized by a strong cross-polarized scattering component. One may note the position of the CSs in the region of the park (in the walking promenades and on the central place within the park) and in the dense urbanized areas (on the corners of buildings, along the streets, and on the two bridges over the Elbe River).

The detected CS are characterized, in general, by high amplitudes, and low polarimetric entropy values. Their majority has a man-made character, a fact that predicts a relative high temporal stability. Because of their strong polarized behavior polarimetric acquisition diversity increases significantly the performance of CSs detection. Indeed for the Dresden data set the number of detected CSs by using fully polarimetric data, and an optimization similar to (21.157), is by a factor up to ten higher compared to the number of CSs detected by using single-polarization data. The fact that the majority of the CSs is not depolarizing, and can be described by its scattering matrix, makes polarimetric information essential for the characterization, interpretation, and information extraction from individual CSs [51].

2.21.4.1.3.4 CS detection based on multiple criteria

In [52], a study of High Resolution X-band images is conducted over urban structures. The stability of the TF responses using a feature vector containing the coefficient of variation of the signal intensity, the TF entropy and gradient-based indicators of the stability of the continuous TF response. A statistical analysis of this feature vector is led to discriminate scatters having a coherent or unstable behavior in frequency, in azimuth or in both directions. A polarimetric analysis is led over specific objects. At such a high resolution, the TF behavior of objects could be used for their recognition using automatic techniques. An illustration is given in Figure 21.60.

2.21.4.1.4.1 Polarimetric Time-Frequency features

Figure 21.61 shows a color-coded polarimetric SAR image of the city of Dresden acquired by DLR’s E-SAR sensor data at L-band. The scene is mainly composed of built-up areas including vegetation spots. A forest and a park can be seen on the left part of the image and a river with smooth banks is located on the right part.

Polarimetric properties of media are generally investigated through a decomposition of second order multivariate polarimetric representations. The resulting parameters provide information on the media geometrical structure and on the underlying scattering mechanisms. Two parameters, obtained from the well known eigenvector-based decomposition introduced in [13] are displayed in Figure 21.62.

The entropy image shown in Figure 21.62 reveals that the polarimetric behavior of most of the scene is highly random. Over urban areas, the polarimetric response is composed of a large number of different polarimetric contributions originating from complex building structures as wall as from surrounding vegetation. The resulting high entropy involves that an interpretation of polarimetric indicators may not be relevant. Over buildings aligned with the flight track direction, the entropy has intermediate values and the parameter reveals the presence of dominant single and double bounce reflexions. Buildings which do not face the radar track are characterized by a strong cross-polarization component and high entropy and can hardly be discriminated from vegetated areas.

Figure 21.63 presents a continuous TF analysis in the azimuth direction of three different media: a building facing the radar, an oriented building, and a forested area. The SPAN, corresponding to the sum of the intensities in all polarimetric channels, and the polarimetric angle are computed for the different media at each frequency location and mean values are then estimated over pixels belonging to the object.

A non stationary behavior is clearly visible in Figure 21.63 with a sudden a large variation of both SPAN and levels with the observation angle in azimuth . This anisotropic behavior is due to the highly directional patterns of coherent scattering mechanisms which may occur as the radar faces a large artificial structure, such as a building [53]. This particular effect can only be observed if the building orientation with respect to the radar flight track falls within the processed antenna azimuth aperture.

On the contrary, oriented buildings and vegetated areas, like forest patches, show stationary behaviors. The identification of buildings from their TF response thus requires an additional criterion to complement the stationarity information. It is known that man-made objects are likely to have a coherent response, whereas natural media may be considered as random. The discrimination of such responses can be achieved by studying the coherence of the backscattered polarimetric signal in the Time-Frequency domain and requires the use of an adequate TF polarimetric SAR (PolSAR) signal model.

2.21.4.1.4.2 PolSAR TF signal modeling and analysis

PolSAR data TF model. The proposed TF signal model [54] is given by the following expression, where the spatial coordinates, , have be omitted:

(21.160)

The signal contains the full coherent polarimetric polarization information and can be associated to a well-known scattering vector [13]:

(21.161)

where represents an element of the scattering matrix sampled at the frequency coordinates .

The signal described in (21.160) is composed of two contributions:

This composite model may be tested using second order statistics:

Second order statistics. Due to the signal high dimensionality, usual scalar tools are not well adapted to the study of second-order TF polarimetric statistics. A polarimetric TF target vector is built by gathering the PolSAR information sampled at spectral coordinates .

(21.162)

The sampling coordinates, , and the frequency domain resolution of the analyzing function are chosen so that the sub-spectra do not overlap and span the whole full resolution spectrum [48]. A polarimetric TF sample covariance matrix, , is then computed as follows

(21.163)

Non-stationary pixel discrimination. Stationarity is assessed by testing the fluctuations of the variance of the signal at the different spectral locations [46,48]. In the polarimetric case, the signal sample variance is given by a polarimetric coherency matrix, i.e., by the diagonal terms of the matrix: . The polarimetric TF response is considered as stationary if the sample matrices, assumed to follow independent complex Wishart distributions with looks, have the same expectation . The corresponding hypothesis is given by:

(21.164)

The corresponding Maximum Likelihood (ML) ratio is:

(21.165)

The likelihood terms in (21.165) are maximized by replacing the expectation matrices by the ML estimates and the hypothesis is tested using the resulting ML test [45], which takes the following form:

(21.166)

Figure 21.64 presents a log-image of the parameter on the Dresden test site, obtained with four spectral coordinates in the azimuth direction over the Dresden test site.

The parameter reach high values over natural areas indicating a stationary spectral behavior. Over buildings, decreases, pointing out the invalidity of the stationary hypothesis over such objects. Highly anisotropic pixels, such as those corresponding to the wall-ground dihedral reflection or specular reflection from oriented roofs are clearly identified in Figure 21.64 due to their very low stationary aspect.

Coherent pixel discrimination. In [51], the eigenvalues of a single-polarization covariance matrix have been used to derive a coherency indicator. These eigenvalues carry information on the correlation structure, but are also sensitive to potential PolSAR fluctuations due to non-stationarity. A solution has been proposed in order to overcome this limitation and to jointly use all the polarimetric channels [54]. Under the hypothesis of uncorrelated spectral responses, the off-diagonal terms of the TF covariance matrix verify:

(21.167)

The corresponding ML ratio is given by:

(21.168)

This ML ratio expression can be rewritten as:

(21.169)

with

(21.170)

The normalized covariance matrix, results from the whitening of the TF polarimetric covariance matrix by the separate polarimetric information at each frequency location. This representation is then insensitive to spectral polarimetric intensity variations and is characterized by its off-diagonal matrices which can be viewed as an extension of the scalar normalized correlation coefficient to the polarimetric case. The ML ratio in (21.168) is a function of the eigenvalues of , which reflect the correlation structure: flat for decorrelated responses (), heterogeneous for correlated ones. Taking into account peculiar form, a correlation indicator, named TF-Pol coherence, can be defined as [54]:

(21.171)

Figure 21.65 presents an image of over the Dresden test site, computed from four spectral locations in the azimuth direction.

As expected, the TF-Pol coherence is high over buildings due to the presence of strong coherent reflectors. It can be also noticed that buildings are identified independently of their orientation.

2.21.4.1.4.3 PolSAR TF analysis

PolSAR TF classification. The stationarity and coherence indicators derived above can be merged to classify the scene. Both and parameters are thresholded and combined into four classes. The application of the fusion strategy over the Dresden site is shown in Figure 21.66.

The resulting map permits a good estimation of building locations. A physical interpretation can be given for each of the four classes:

TF cleaning of PolSAR data. As it was shown in Figure 21.62, the full resolution PolSAR information can hardly be used to analyze the scene geophysical properties due to a very high entropy inherent to the study of dense environments. The proposed PolSAR TF analysis technique can also be used to improve in a significant way the interpretation of polarimetric indicators. The most coherent TF scattering mechanisms is described by the first eigenvector of , which can be transformed back to the H-V polarimetric basis using a matrix , satisfying . From this eigenvector, one can extract an parameter which shows a much more contrasted and relevant information than the original full resolution parameter . Figure 21.67 shows different images of a building of the scene. A comparison between the T-F classification results and the optical image reveals that both the double bounce reflexion is considered as a non-stationary and coherent scattering mechanism, whereas the roof layover is seen as non-stationary and uncorrelated, due to the superposition of the roof and ground contributions. A thresholding of with respect to , permits to easily separate these two different mechanisms and could be used to get a rough estimate of the building height. Such an information might be useful to interferometric phase unwrapping algorithms which generally face ambiguity issues over urban areas.

2.21.4.2.1.1 Relation between interferometric phase and the height of a scatterer

The geometrical configuration of an interferometric acquisition, depicted in Figure 21.68, shows two sensors, located at slightly different positions, that measure the coherent SAR response of a scatterer. The relative positions of the sensors is measured by the baseline and the angle . After focusing of the SAR signals and co-registration of the SAR images, the response of the scatterer is given by

(21.172)

where represents the absolute and unknown phase of the reponse of each object. For a small baseline and if the acquisitions are made within a sufficiently short amount of time, the amplitude and phase of the object responses may be considered identical, . In this case, the interferometric phase difference, or interferogram, is given by

(21.173)

This interferometric phase being proportional to the path difference, , it is sensitive to both the elevation of the scatterer with respect to a reference plane and to the range distance . A local linearization permits to isolate the elevation-dependent part :

(21.174)

where can be computed from the geometrical configuration and hence compensated and is given by

(21.175)

One may note that the estimation of the interferometric phase as the argument of a complex exponential may imply a wrapping due to the limited domain of definition of the operator. As a consequence

(21.176)

This ambiguity can generally be overcome using phase unwrapping techniques.

2.21.4.2.1.2 Interferometric coherence

Over homogeneous distributed environments, the acquired are random variables whose statistics may be studied using the 2-element vector, , following a complex normal distribution, , where the covariance matrix may be written as

(21.177)

where

(21.178)

is called the interferometric coherence and may be used to estimate the interferometric phase . Using independent realizations, or looks, of , ML estimates may be computed as [21,22]

(21.179)

In addition to the need of a large number of looks, , a good estimate of the interferometric phase requires a large value of the modulus of the coherence, since when and reaches a maximum as .

The coherence may be modeled as a product of a large number of contributions, as, for instance:

(21.180)

Each element represent a potential source of decorrelation between the acquired signals, due to the presence of additive noise, errors during the acquisition, processing and referencing of the signals, to changes of the scenes between the acquisitions, to spectral shifts induced by acquisition geometry and to vertical decorrelation induced by the presence of a volumic medium. Very few of these can be compensated a posteriori and each of them may influence the phase estimation accuracy since

(21.181)

2.21.4.2.1.3 The random volume over ground model for a single polarization channel

The last term of the coherence decomposition in (21.180) is related to the structure of the observed medium through an integral relation [15,55–58]

(21.182)

where represents the equivalent density of coherent reflectivity in the elevation direction. The interferometric coherence may then be used to guess some characteristics of the vertical structure of the observed volume, provided that the other terms in (21.180) are known or compensated. In order to estimate the possibilities of such an approach, one may use a simple model of a homogeneous Random semi-transparent Volume lying over a Ground consisting of a rough surface, the RVOG model, depicted in Figure 21.69[56,58].

The volume is assumed to be made of particles having the same reflectivity and the same constant extinction properties. i.e.,

(21.183)

then

(21.184)

where the factor account for round-trip attenuation. The ground reflectivity distribution is then given by

(21.185)

The intensities backscattered by the ground and the volume are given by

(21.186)

The general unresolved expression of the coherence is then

(21.187)

In order to provide a physical interpretation for expression (21.187), cases with different levels of complexity can be investigated.

Case 0: Transparent volume. In this extremely simple case, the volume response is null, i.e., . The coherence

(21.188)

has a unitary modulus and its argument provides the elevation of the ground.

Case 1: Lossless volume and no ground. The extinction of waves when they travel through the volume is null, and the ground reflectivity is null, . In this case represents the coherence behavior of the volume only

(21.189)

The coherence argument indicates exactly the absolute elevation of the center of the volume, whereas its modulus depends on its width . This case is illustrated in Figure 21.70.

Case 2: Volume with constant linear extinction and no ground. The extinction is constant over the volume and the ground reflectivity is null, . The expression of the coherence is

(21.190)

The magnitude and phase behaviors of are illustrated in Figure 21.71 for different values of extinction. For a quasi-null extinction, the coherence magnitude has a sinc-like shape and the phase center is located at the center of the volume. As the extinction increases, the magnitude pattern becomes flatter and close to when is large. The phase center moves then towards the higher interface of the volume. This behavior is due to the fact that as increases the wave penetration depth inside the volume decreases and the particles that participate more to the scattering phenomenon are those located near the upper interface. For reduced penetration depths, the decorrelation induced by the numerator of (21.190) becomes negligible.

Case 3: Volume with constant linear extinction and ground. The extinction is constant over the volume and the ground reflectivity is non null. The expression of the coherence is [55,56,58]

(21.191)

where , called the ground to volume intensity ratio, can be considered as a key parameter of the RVOG coherence which steers the coherence between to extreme behaviors, for and . For intermediate values a single coherence term is not sufficient for analyzing the RVOG model.

2.21.4.2.4.1 Vegetation bias removal

In practice the ground to volume intensity ratio never reaches the limits of its domain of definition , i.e., . For a ground contribution with a sufficiently low entropy, there exist a scattering mechanism, typically HV, so that and in this case . Due to the lack of polarization selectivity of volume scattering, the extremity of the line segment corresponding to cannot be used to estimate the ground elevation. Cloude and Papathanassiou proposed to overcome this problem by extrapolating the line segment until it intercepts the unitary circle, since the RVOG model for a transparent medium, . This physical interpretation permits to reduce an important elevation bias linked with the completely depolarized volume response.

2.21.4.2.4.2 Practical vegetation height estimation

In [55], Cloude proposes a theoretically sub-optimal but practically robust method to estimate the height of a volume from POL-inSAR data sets.

Line segment. From a L-look sample Pol-inSAR coherency matrix, estimate the line parameters, e.g., and in Figure 21.73. It is recommended to use the analytical expressions provided in [59] instead of a least-square fit based on sample coherences generated from (21.195) and a set of pre-defined or random projection vectors. Indeed the approach in [59] is immediate and corresponds to the generation of an infinite number of coherences.

Volume and ground coherences. This can be done by selecting one of the segment extremity and as , e.g., one may chose the one closer to . Extrapolate the other extremity of the segment to the unitary circle in order to determine .

Estimate the height assuming an infinite extinction. As mentioned earlier, for an infinite extinction value, the phase center of correspond to the upper interface of the volume. Then

(21.199)

It is unlikely that , so will indeed correspond to a phase center located within the volume. This value might underestimate the true volume height.

Estimate the height assuming an null extinction. For a null extinction, the volume coherence magnitude as a sinc-like shape. The corresponding height might be estimated as

(21.200)

It is also unlikely that , i.e., that the phase center is at its lowest possible elevation. This value might underestimate the true volume height.

Empirical linear combination. The final height estimate is

(21.201)

Recommended value: 0.4

This semi-empirical approach reveals more robust than a theoretically more correct one, since it combines uses separate estimates based on the modulus and phase of , linearly combines estimates with opposed biases and does not use numerical values of the extinction that main cause severe artifacts. Examples of results obtained over forested areas using polarimetric and interferometric SAR processing are given in Figures 21.74 and 21.75.

2.21.4.3.2.1 TomSAR signal models

Considering an azimuth-range resolution cell that contains backscattering contributions from scatterers located at different heights and assuming no decorrelation between the different acquisitions, the received data vector, , can be formulated as follows:

(21.204)

where indicates one of the independent realizations of the signal acquisition, also called looks. The source signal vector, , contains the unknown complex reflection coefficient of the scatterers, and represents the complex additive noise, assumed to be Gaussianly distributed with zero mean variance , and to be white in time and space with, i.e., and . The steering vector contains the interferometric phase information associated to a source located at the elevation position above the reference focusing plane and is given by:

(21.205)

where is the two-way vertical wavenumber between the master and the th acquisition tracks. The corresponding perpendicular baseline is aligned with the cross-range direction. The carrier wavelength is represented by , whereas stands for the incidence angle and is the slant range distance between the master track and the scatterer. The steering matrix consists of steering vectors corresponding each to a backscattering source:

(21.206)

with , the vector of unknown source heights.

Considering now interferometric decorrelation between different acquisitions, the initial model in (21.204) may be reformulated as a sum of contributions from random sources [60]:

(21.207)

where accounts for both the reflection coefficient of the th source, and its potential variations between the interferometric acquisitions or over the realizations. Depending on the type of scatterer under observation, the source signal possesses varying statistical properties. The resulting composite signal may then follow different scattering behaviors [66].

2.21.4.3.2.2 PolTomSAR signal model

The polarimetric response of a scatterer is fully described by its () complex scattering matrix . The scattering matrix can be vectorized using, for instance, the Pauli basis matrix set [12], in order to build a target vector

(21.211)

where represents the polarimetric span [12] of the scatterer response, and represents a unitary polarimetric target vector, i.e., . In an MB-PolInSAR configuration, the array response may be represented by re-arranging the acquired polarimetric signals, with the track index, under the form of a element vector, , composed of 3 MB-inSAR components, each related to a polarization channel:

(21.212)

where , with , represents the MB-InSAR response for the th polarimetric channel, i.e., . Using this convention of representation, the polarimetric steering vector and steering matrix are given by:

(21.213)

where is the polarimetric target vector of the th source and . A polarimetric steering vector can be defined using five real coefficients given by the elevation, and the real and imaginary parts of two complex numbers defining a unitary 3-element polarimetric complex vector whose absolute phase is arbitrary. Similarly to the Single Polarization (SP) expression given in (21.204), the received MB-PolInSAR signal may be formulated as

(21.214)

where similarly to the SP case, represents a realization of the complex amplitude of the th source. Diverse model assumptions given for SP signals, can be similarly used for PolTomSAR signals.

2.21.4.3.3.1 Single polarization tomography

Using MB-InSAR data sets acquired for a given polarization channel, Single Polarization (SP) tomography can be derived from the data covariance matrix using classical mono-dimensional estimators like Beamforming, Capon and MUSIC estimator as well as multi-dimensional methods like maximum likelihood estimators and weighted subspace fitting approaches.

Mono-dimensional estimators. These approaches determine , an estimate of the elevation of the scatterers under observation, as the coordinates of the largest local maxima of an continuous objective function :

(21.217)

For the classical Beamformer and Capon spectral estimation techniques, the objective function is given by the continuous estimate of the reflectivity, , defined as

(21.218)

Once the set of scatterer elevation, , is estimated using (21.217), the corresponding reflectivities can be obtained from (21.218). The selection of discrete sources from peaks of the reflectivity spectrum confers to the Beamformer and Capon estimation techniques an important sensitivity to the acquisition configuration, and in particular to the presence of spurious sidelobes related to an irregular baseline sampling. The Beamformer is known to show a low resolution and may then overlook some closely spaced scatterers, whereas Capon’s technique possesses an improved resolution but a reduced radiometric accuracy.

MUSIC is a subspace based mono-dimensional technique, whose objective function is a measure of the orthogonality between a steering vector and the estimated noise subspace and is given by:

(21.219)

Once , is determined by inserting (21.219) in (21.217), a Least-Square (LS) estimate of the complex reflectivity vector can be obtained [60,61].

Nonparametric approaches like Beamforming and Capon, are generally used to globally appreciate the structure of a volumetric medium and the main trends of the continuous reflectivity distribution in elevation. For the analysis of discrete spectral components, they may fail to discriminate closely spaced scatterers due either to their limited resolution, or to the presence of side lobes that may induce an erroneous estimation of the source location. MUSIC generally presents better performances for the analysis of discrete sources, related to a better resolution. Nevertheless, like all parametric methods, MUSIC is sensitive to data modeling errors, and in particular those related to the estimated number of sources . Moreover, MUSIC is known to work well in the case of uncorrelated scatterers, but its performance may degrade significantly in the presence of correlated scatterers since the source signal covariance matrix tends to be singular [72]. One of the main advantages of such techniques resides in the low numerical complexity of the mono-dimensional optimization described in (21.217).

Multi-dimensional estimators. Maximum likelihood (ML) estimators are multi-dimensional techniques, generally given by

(21.220)

with the likelihood function . In the case of distributed scatterers, the Stochastic ML (SML) can be derived [73]:

(21.221)

where represents the orthogonal projector on the null space of . Considering deterministic scatterers, the cost function (21.220) results in the Deterministic ML (DML) estimator[74]

(21.222)

Weighted Noise subspace fitting (NSF) estimator is derived by fitting the estimated noise subspace with in the weighted LS sense as follows:

(21.223)

whereas the weighted Signal Subspace Fitting (SSF) cost function is given by:

(21.224)

where the fitting matrix is replaced by its LS estimate . Both cost functions may be used to estimate the elevation of the scatters using a -dimensional minimization:

(21.225)

where the suffix WSF indicates ones of the methods, NSF or SSF, mentioned above. It has been shown in [75], that any hermitian positive semi definite weighting matrix yields consistent parameter estimates. In particular, a consistent estimate of permits to obtain minimum variance estimates, which asymptotically reach the Cramr-Rao lower bound [73]. Such a value is given for each case:

(21.226)

Compared with the mono-dimensional ones, multi-dimensional methods ML estimators are more robust, leading to global optima, but at an expensive computational cost. Regarding weighted subspace fitting techniques, the appropriate selection of certain weighting matrices can provide these estimators an optimal estimation accuracy as ML techniques at a reduced computational cost.

Numerical examples. In order to demonstrate the performance of the aforementioned tomographic estimators, two scatterers are assumed to locate at and , acquired respectively by a multibaseline configuration with acquisitions. The baselines are assumed to be evenly distributed and between each successive acquisitions, the difference of vertical wavenumbers is supposed to be . Assuming these two scatterers with equal reflectivity, the covariance matrix of source signals is given by

and . Varying the correlation factor can respectively represent the source covariance matrix of coherent (), distributed (), and hybrid scatterers (). In order to study the vertical resolution, these estimators are used in the height estimation of uncorrelated signals () with varying the height difference of two scatterers . Figure 21.79a shows when , MUSIC estimator degrades significantly due to closely spaced scatterers. SSF and NSF estimators both provide very good resolution with . Whereas, Capon’s method can provide a good height estimate when . Now simulating a 256-look sample data covariance matrix with varying from to , RMSE of the estimated is plotted with respect to the correlation factor in Figure 21.79b. The NSF estimator provides the most accurate estimate for uncorrelated or partially correlated signals (), while the SSF estimator copes well with highly correlated signals (). MUSIC cannot deal with highly correlated sources due to the quasi-singularity of .

2.21.4.3.3.2 Fully polarimetric tomography

Fully Polarimetric (FP) tomography is implemented by jointly using fully polarimetric data sets acquired in a Polarimetric MB-InSAR configuration. It is a useful tool to extract both the scatterer’s vertical location and its scattering mechanism. Similar to the SP case, the FP tomography is derived from the data covariance matrix using some mono-dimensional and multi-dimensional FP spectral estimators.

Mono-dimensional estimators. The elevation and scattering mechanisms of the different scatterers are estimated as the coordinates of the largest local maxima of a polarimetric objective function .

(21.227)

Similarly to the SP case, the Beamformer and Capon objective functions are given by continuous estimates of the reflectivity , with

(21.228)

The direct local maximization of the FP objective function in (21.227) would require searching solutions in a 5-D argument space. Rewriting the polarimetric steering vector as

(21.229)

permits to formulate reduce the search dimension to one, since

(21.230)

where represents the minimum and maximum eigenvalue of the positive semi-definite matrix . Once the elevations, are determined from the local maxima of (21.230), the corresponding scattering mechanisms can be estimated from:

(21.231)

where indicates the eigenvector of associated with , respectively. A similar approach may be applied to concentrate the FP MUSIC objective function:

(21.232)

where represent here the eigenvectors of spanning its noise space. The reflectivity of the sources can be estimated, using a L-look polarimetric LS approach.

These estimators are considered as mono-dimensional, since the estimation of the elevation and the scattering mechanism is jointly realized by eigendecomposition. Such methods are computationally efficient but may reach some of the limitations mentioned in the SP case.

Multi-dimensional estimators. The ML estimators are formulated in the FP case using the polarimetric steering matrix , represented by:

To directly optimize the FP ML estimators, the estimation of optimal elevations and target vectors , requires an optimization over a -dimensional space, and implies an excessive computational burden.

Polarimetric WSF estimators are given by:

Directly minimizing the FP-SSF estimator also suffers computational cost, but the FP-NSF estimator can maintain the computational cost as the SP case, using an analytic solution proposed in [66]. Compared with SP tomography, polarimetric tomography can provide more accurate localization of scatterers in the vertical direction due to the polarization diversity. The observed media can be further characterized in terms of both vertical location and scattering mechanism.

Numerical examples. Under the same baseline configuration as the SP case in Section (2.21.4.3.3.1), a 256-look sample covariance matrix is derived from received fully polarimetric data sets in the MB-PolInSAR configuration, which contains both height and polarimetric information. Assuming the above two scatterers with the canonical surface scattering, i.e., , Figure 21.80a shows RMSE of height estimation w.r.t. : FP-Capon estimator still cannot accurately separate two scatterers due to a limited resolution; FP-NSF estimator provides the most precise height estimate; FP-MUSIC estimator degrades its performance for highly correlated scatterers. To study the role that polarization diversity plays in source separation, we denote Polarization Difference Angle with . We keep the same surface scattering for the scatterer at 0 m and vary the scattering mechanism of the scatterer at 4 m to increase the polarization difference angle. As shown in Figure 21.80b, with increasing, polarization diversity between two scatterers improves the height resolution, especially for FP-Capon estimator which reaches the same resolution as MUSIC when . FP-NSF estimator performs best over all the polarization difference angle.

2.21.4.3.4.1 Urban remote sensing

High-resolution (HR) tomographic techniques are applied to analyze dense urban areas in terms of scatterers’ heights and their scattering mechanisms. Diverse scattering patterns can be encountered over dense urban environments, such as double bounce reflection at the wall-ground interaction, surface scattering from the roof and the ground, volumic scattering or a mixture of several scattering patterns. Conventional tomographic estimators may reach some limitations. The proposed model-adaptive FP-NSF estimator in [66,76] illustrates an undeniable performance using L-band intermediate-resolution dualbaseline FP data sets acquired over Dresden, Germany, by the DLR ESAR sensor. Over a test line, the tomograms obtained by FP-MUSIC and FP-NSF estimators with model order equal to 2 are shown in Figure 21.81. The heights of buildings are validated against LIDAR data as shown in [77]. FP-MUSIC estimator suffers spurious sidelobes (blue¹ circle), whereas FP-NSF estimator illustrates precisely the vertical profile of buildings without any sidelobe. The 3-D reconstructions of an urban zone are depicted in Figure 21.82.

2.21.4.3.4.2 Forestry remote sensing

In the forest scenario, the canopy consists of a large number of elementary scatterers distributed in the vertical direction and is characterized by a continuous spectrum, whereas the ground is an impenetrable medium that possesses an isolated localized phase center and is characterized by a discrete spectrum. So the vertical distribution of backscattered power from forested areas leads to a mixed spectrum. The performance of conventional tomographic estimators may degrade due to their lack of adaptation to the type of spectrum. A novel hybrid tomographic approach is proposed to deal with the mixed spectrum and provides robust estimates for both tree top heights and the underlying ground topography [77].

Using P-band data sets (including 6 tracks) acquired over a test area of Paracou, French Guiana, by the ONERA SETHI sensor, the tree top and the ground elevation over an azimuth cut are estimated by the hybrid spectral approach (in gray) and the corresponding Lidar profiles (in black) are drawn over the normalized Capon tomograms in Figure 21.83. It can be observed that the estimated profiles coincide well with the Lidar ones. The ground elevation is precisely estimated using HH data sets, but still overestimated in VV and HV polarization. The forest top heights are well-estimated over all polarization channels. Using FP data sets, the results are similar to those obtained by HH data. Using HH data sets, Figure 21.84 depicts the estimated and over a test zone as well as the corresponding Lidar data in ground range. Globally, the estimated and match well with the Lidar ones in terms of texture and heights, except for some overestimated areas due to topographic effects that may influence scattering responses.

2.21.4.3.4.3 Underfoliage imaging

For the study of under-foliage imaging, Capon’s method has been applied to extract the shape of objects and canopy profile [78], but considering the complex structure of such objects, HR approaches are required to discriminate their closely spaced scattering features in the vertical direction. The performance of these spectral analysis techniques is conditioned by the statistical nature of the scattering response of the observed objects, as it has been shown in [79,80]. Under-foliage objects may be described by a deterministic response embedded within a speckle affected environment and hence associated to a series of complex scattering centers. To distinguish under-foliage objects, Figure 21.85 illustrates the tomograms over a test line with a truck beneath the canopy [81], using L-band PolInSAR data sets acquired over Dornstetten, Germany. The 3-D tomogram of the area, performed over azimuth bins and for varying range positions using the SP-SSF approach. Limiting the reconstruction height to 4 m above the terrain permits to isolate the under-foliage truck response (Figure 21.86a). Its reconstructed shape is similar to the uncovered one, with a reflectivity slightly higher (about 5 dB more) than the surrounding environment one. Complementary to distinct the objects in reflectivity, a 3-D reconstruction using FP-NSF estimator, permits to visualize the corresponding scattering pattern by value. To better visualize the underlying truck response, the height is limited to 4 m above the ground and the reflectivity is limited to values inferior to 43 dB (Figure 21.86b). It shows that the truck outside the forest has a strong double bounce reflection at the ground-truck interaction, and the shape of the truck beneath the canopy is a precisely reconstructed with an value about due to the sidelobe effect from the canopy. Using PolTomSAR techniques, the under-foliage truck can be well described both in terms of shape and scattering pattern.