Improved Coherent Processing of Synthetic Aperture Radar Data through Speckle Whitening of Single-Look Complex Images ^†

Abstract

In this study, we investigate the usefulness of the spectral whitening procedure, devised by one of the authors as a preprocessing stage of envelope-detected single-look synthetic aperture radar (SAR) images, in application contexts where phase information is relevant. In the first experiment, each of the raw datasets of an interferometric pair of COSMO-SkyMed images, representing industrial buildings amidst vegetated areas, was individually (1) synthesized by the SAR processor without Fourier-domain Hamming windowing; (2) synthesized with Hamming windowing, used to improve the focalization of targets, with the drawback of spatially correlating speckle; and (3) processed for the whitening of complex speckle, using the data obtained in (2). The interferograms were produced in the three cases, and interferometric coherence and phase maps were calculated through 3 × 3 boxcar filtering. In (1), coherence is low on vegetation; the presence of high sidelobes in the system’s point-spread function (PSF) causes the spread of areas featuring high backscattering. In (2), point targets and buildings are better defined, thanks to the sidelobe suppression achieved by the frequency windowing, but the background coherence is abnormally increased because of the spatial correlation introduced by the Hamming window. Case (3) is the most favorable because the whitening operation results in low coherence in vegetation and high coherence in buildings, where the effects of windowing are preserved. An analysis of the phase map reveals that (3) is likely to be facilitated also in terms of unwrapping. Results are presented on a TerraSAR-X/TanDEM-X (TSX-TDX) image pair by processing the interferograms of original and whitened data using a non-local filter. The main results are as follows: (1) with autocorrelated speckle, the estimation error of coherence may attain 16% and inversely depends on the heterogeneity of the scene; and (2) the cleanness and accuracy of the phase are increased by the preliminary whitening stage, as witnessed by the number of residues, reduced by 24%. Benefits are also expected not only for differential InSAR (DInSAR) but also for any coherent analysis and processing carried out performed on SLC data.

1. Introduction

Thanks to their all-weather and nighttime operational capabilities, synthetic aperture radar (SAR) satellite systems are very attractive and thus are widely used for a number of applications concerning environmental monitoring, frequently increasingly in synergy with artificial intelligence tools [1]. The fusion of SAR and optical data constitutes a unique chance to merge the complementary abilities of the two types of data [2,3]. SAR images, however, are corrupted by speckle, which is a signal-dependent noise peculiar to all coherent imaging systems that severely impairs the appearance of images and drastically reduces the performance of automated analysis tools. Fully developed speckle is modeled in the spatial frequency domain as a white process. However, in the complex spatial domain of in-phase and quadrature components, speckle is modeled as a spatially uncorrelated, multiplicative random process, independent of the underlying radar reflectivity, described by a zero-mean circular Gaussian probability density function (PDF) [4]. However, when the raw SAR data are processed to synthesize an image, the validity of this assumption may be violated. Before the SAR processor, the raw data are slightly oversampled and weighted in the Fourier domain, where the deconvolution process of the system’s point-spread function (PSF) is performed, to avoid two-dimensional Gibbs effects around bright targets. As a consequence of the use of frequency windows (mostly Hamming or Kaiser) along both range and azimuth, speckle becomes autocorrelated. Hence, the single-look complex (SLC) image is synthesized with speckle that is spatially correlated. This problem was first noticed by some authors who adjusted their post-processing in order to tackle correlated speckle [5,6], and later by many others, e.g., [7,8]. Although incoherent multilooking is usually performed, when phase information is unnecessary, the reduced spatial correlation achieved through multilooking implies a degraded spatial resolution; such a loss of information may not be tolerated in several application contexts.

SAR interferometry (InSAR) exploits the phase differences of two or more complex-valued images relative to the same geographic area and taken from slightly different positions and/or at different times [9]. The information extracted from the phase difference is utilized for the estimation of topography [10] and the measurement of millimetric ground deformations (subsidence) for environmental risk monitoring and security applications. Also, analyses of ocean surface currents benefit from an interferometric approach [11]. Angular, or baseline, decorrelation, induced by spatial diversity of orbits, and temporal decorrelation, induced by changes in the dielectric properties of the scene, are the main factors that influence the dependability of the interferometric phase. Point targets and strongly textured regions play a fundamental role, as their distribution of elementary scatterers does not change over time. In contrast, areas covered by vegetation may exhibit temporal decorrelation, which impairs the measurements of the interferometric phase [12]. Coherent spatial multilooking of the complex interferogram is generally utilized to counteract the decorrelation effects induced by scattering mechanisms affecting a wide number of pixels. The interferometric coherence, estimated through the coherent multilooking of the normalized interferogram, is inversely related to the noise of the interferometric phase [13,14]. When coherence is equal to one, speckle patterns in the two images are identical, and the interferometric phase is noise-free.

The transfer function of the SAR system in the spatial frequency domain, referred to as the spatial frequency response (hereafter, the frequency response), may induce a spatial correlation of the speckle if frequency-domain filtering, e.g., achieved through a Hamming window, is introduced in the SAR processor that focuses the raw data into an SLC image. The drawback of achieving focused targets is that speckle becomes spatially correlated in either range, azimuth, or both, depending on where the window is applied. While geometrically accurate targets are desirable, the autocorrelation function of speckle may induce an artificial increment of coherence in areas, mainly vegetated, affected by a temporal decorrelation of radar echoes. The coherence bias mainly depends on the missing local ergodicity of the data, originating from autocorrelation. High coherence values are the prerequisite for an accurate unwrapping of the phase field; thus, the coherence bias may result in erroneous detection of areas in which the phase is reliable and may be unwrapped. Although the frequency window may be enabled or disabled upon request, the two benefits of well-focused targets and spatially uncorrelated speckle cannot be jointly achieved.

Coherent processing of temporal sequences of SLC images includes sea surface complex wind field estimations [15], measurements of interferometric coherence over time, e.g., to analyze changes in glaciers and snow cover [16], or rainforest mapping [17]. Generally speaking, temporal coherence analysis provides a valuable tool for investigating subtle variations in land cover, independent of the dielectric and geometric properties of the surface, and is related to the temporal correlation of detected speckle patterns in a sequence of images [18]. In fact, high coherence indicates not only that the reflectivity of the surface is unchanged but also that the spatial distribution of elementary scatterers within the resolution cell, approximately twice the pixel size, is unchanged.

Another possible application of deconvolution for speckle whitening is in multi-baseline SAR tomography, stemming from interferometry. In this emerging technique, a stack of SLC images, acquired from slightly different antenna phase center positions in the vertical plane that constitute a cross-track array [10,19], is processed in the baseline domain to obtain a synthetic beam in the elevation dimension to resolve the scattering distribution in full 3D [19,20]. The most important current scenario for SAR tomography is the sensing and monitoring of forest structures and biomass, which is of great importance to the carbon cycle and global warming issues [20]. Benefits are expected from whitened speckle in tomographic processing, as well as in interferogram estimation, due to the increased number of effective looks [19,20].

Exactly one decade ago, a viable strategy for “whitening” spatially correlated speckle in SLC data, without prior knowledge of the SAR system and the raw-data processor, was developed by one of the authors [6]. Spatially correlated speckle is perhaps the major reason for the mediocre despeckling performance achieved by advanced methods when single-look amplitude/intensity images [21] are processed for speckle reduction [22]. The procedure in [6] is fully unsupervised and carries out estimation and inversion of the frequency response of the SAR system, with preliminary detection based on percentile thresholding and separate processing of bright targets. In subsequent studies, the applicability of the whitening procedure to polarimetric speckle reduction [23] and polarimetric feature extraction [24] was addressed. Recently, it has been demonstrated that incoherent change detection from SAR images also takes significant advantage of the uncorrelatedness of speckle [25], even for methods that are slightly sensitive to speckle [26]. In this paper, we resume our earlier investigations [27] to determine whether the original procedure [6] can be used for coherent processing, where phase information is invaluable. This includes mainly InSAR but, in principle, also differential InSAR (DInSAR), and coherent change detection based on temporal coherence analysis [28]. Also, some suggestions for its possible application to SAR tomography [19] will be briefly discussed.

The organization of this article is as follows. Section 2 briefly reviews the SAR system to explain the origin of the autocorrelation of speckle and the ways it can be mitigated in practical application contexts. Additionally, the basics of SAR interferometry are presented, and methods for estimating SAR interferograms are discussed. Section 3 presents the quantitative results from two different datasets: one from the COSMO-SkyMed satellite constellation and one from the TerraSAR-X/Tandem-X twin satellites. Section 4 discusses the advantages of spatial decorrelation for InSAR processing and the importance of coherence analysis. Concluding remarks and possible developments are presented in Section 5.

2. Materials and Methods

2.1. Speckle Whitening of Complex SAR Data

2.1.1. SAR System: An Overview

SAR images are computed images synthesized by processing radar returns captured from a moving platform [9]. A train of pulses, also known as chirps, is transmitted along a leaning direction, referred to as the slant range, orthogonal to the track of the platform, which constitutes the conjugate azimuth direction. The SAR system receives and stores the complex returns (amplitude and phase) of the objects encountered in the cross-track direction, which are located, thanks to the delays of their responses, through Doppler processing [29]. In-phase and quadrature components of the complex envelope of the received signal are sampled at a frequency at least twice the bandwidth of the chirps, quantized, and stored; they constitute the two-dimensional (2D) array of raw data.

The 2D array of raw data, which exhibits a grainy, noise-like appearance and a mean that slowly varies in space, has to be focused by the second segment of the SAR system, called the SAR processor, which is usually located on the ground because of its high processing power requirements. SAR processing consists of a 2D deconvolution, typically separable, of the shape of the band-limited pulse [30] along the slant-range direction (range focusing), and of the so-called azimuth chirp, caused by the varying Doppler shift within the antenna beamwidth in the azimuth direction, which develops along the track of the platform and whose shape is squeezed with the range (azimuth focusing) [9,29]. Fourier-domain deconvolution is fast and accurate but cannot be used for drones, whose trajectory is unpredictable [31].

The range and azimuth deconvolutions are intended to emulate (a) a compressed range pulse, which is exactly band-limited and thus has a perfectly rectangular Fourier transform; and (b) a synthetic antenna beam with a narrow main lobe in the azimuth direction. Unfortunately, after the 2D deconvolution, the range chirp becomes a sinc function, that is, the inverse Fourier transform of a rect function; the sinc function has tails that decay slowly along the range. Similarly, in the azimuth direction, after the azimuth chirp has been deconvolved, the same issue occurs. The undesirable consequence of the deconvolution process is that a bright point target becomes surrounded by a dashed cross, not exactly square unless the squint effect [29] is corrected, caused by the 2D Gibbs effect.

The on-ground SAR processor usually employs frequency windows, typically Hamming or Kaiser windows, to avoid the dispersion of point targets. The drawback is that fully developed speckle, which, in principle, should be spatially uncorrelated if the raw data are not oversampled, becomes autocorrelated and presents a grainy appearance. Speckle stems from the coherent combination of radar returns from multiple elementary scatterers at the receiver. If the cell contains few dominant scattering elements, speckle is said to be partially developed. This happens in textures typically found in urban areas [32]. The autocorrelation function of speckle may impact the subsequent analysis/processing of complex SAR data.

Figure 1 shows the end-to-end flowchart of the system, including the (optional) spatial decorrelation stage. The cascade of onboard SAR sensors and the on-ground SAR processor make up the complete SAR system. In principle, the SAR system should yield the product of radar reflectivity and complex speckle that is found at its input. In practice, the SAR processor may include tapering windows along both the range and azimuth directions; hence, the output may become autocorrelated. However, it can be decorrelated by applying the inverse filter of the tapering window.

2.1.2. Problem Formulation

In the following equation, we utilize the properties of SLC SAR images [33]. The observed scene is composed of a set of point scatterers. Let $S^w\left(\mathbf{r}\right)$ denote the discrete complex scattering coefficient and let $\mathbf{r}\triangleq (r_x,r_y)$ denote the spatial 2D coordinate. Under the hypothesis of fully developed speckle, the term $S^w\left(\mathbf{r}\right)$ is a zero-mean, white symmetric circular Gaussian complex process, with variance $\sigma \left(\mathbf{r}\right)$, representing the radar reflectivity measured by the SAR system [29]. The complex-valued image found at the output of the SAR processor, equipped with a 2D separable tapering window, $H\left(\mathbf{f}\right)$, with $\mathbf{f}\triangleq (f_x,f_y)$, can be regarded as the convolution of the discrete scattering coefficient with the PSF of the SAR system, $h\left(\mathbf{r}\right)$, which is the inverse Fourier transform of the tapering window, $H\left(\mathbf{f}\right)$:

$$S\left(\mathbf{r}\right)=S^w\left(\mathbf{r}\right)\ast h\left(\mathbf{r}\right)$$

(1)

which can be expressed in the Fourier domain as

$$S\left(\mathbf{r}\right)={\mathfrak{F}}^{-1}\left\{\mathfrak{F}\left\{S^w\left(\mathbf{r}\right)\right\}·H\left(\mathbf{f}\right)\right\}.$$

(2)

In Equations (1) and (2), we assume that the complete SAR system, including onboard acquisition and on-ground focusing, is modeled as a cascade of linear shift-invariant filters, defined by their PSFs.

The SAR image with fully developed speckle is given by:

$${\left|S\left(\mathbf{r}\right)\right|}^2=\sigma \left(\mathbf{r}\right)·u_u\left(\mathbf{r}\right)$$

(3)

where $u_u\left(\mathit{r}\right)$ is the fading term, usually modeled as an uncorrelated stochastic process with both mean and variance equal to one and a negative exponential PDF. Equation (3) describes the intensity format of a single-look image, i.e., the power of the backscattered signal. By definition [34], the signal-to-noise ratio (SNR) of a single-look SAR image in intensity format is zero dB. The model in Equation (3), however, is valid if the overall SAR system in Figure 1 features a PSF that is a discrete $\delta \left(\mathit{r}\right)$ function. The case of polarimetric SAR (polSAR) is much more complex because speckles are not independent among the polarimetric channels [35], even though each channel is processed independently of the others.

A more general formulation of the problem, including an autocorrelation function of the fading term, can be stated as

$${\left|S\left(\mathbf{r}\right)\right|}^2\approx \sigma \left(\mathbf{r}\right)·u_s\left(\mathbf{r}\right)$$

(4)

where $u_s\left(\mathit{r}\right)$ denotes the fading term, which is spatially correlated due to the nonideal frequency response of the SAR system and is independent of the reflectivity term $\sigma \left(\mathit{r}\right)$. According to [33], this approximation improves as the Fourier transform of the PSF of the SAR system extends further than the power spectrum of the reflectivity. Specifically, the correlation length of speckle should not be greater than that of the reflectivity. Equation (4) accounts for the spatial correlation of speckle, provided that the bandwidth of the system encompasses the power spectrum of the reflectivity.

The goal of the spatial decorrelation of speckle is the removal of the effects of the frequency response of the SAR system to recover $S^w\left(\mathit{r}\right)$ from $S\left(\mathit{r}\right)$. Such a blind deconvolution [36] may be simplified if we assume that the frequency response of the SAR system is a band-limited lowpass filter with a cutoff frequency ${\mathbf{f}}_c$:

$$H\left(\mathbf{f}\right)\approx 0\forall \left|\mathbf{f}\right|>|{\mathbf{f}}_c|.$$

(5)

No assumptions on $\sigma \left(\mathit{r}\right)$ are necessary. In [6], it was proven that an estimate of ${\widehat{S}}^w\left(\mathit{r}\right)$ is given by:

$${\widehat{S}}^w\left(\mathbf{r}\right)=\left\{\begin{array}{ccc}{\mathfrak{F}}^{-1}\{\mathfrak{F}\left\{S\left(\mathbf{r}\right)\right\}·{\left[\widehat{H}\left(\mathbf{f}\right)\right]}^{-1}\}& & \forall \left|\mathbf{f}\right|\le |{\mathbf{f}}_c|\\ \\ 0& & \mathrm{otherwise}\end{array}\right.$$

(6)

where $\widehat{H}\left(\mathbf{f}\right)$ denotes an estimate of the otherwise unknown true frequency response, $H\left(\mathbf{f}\right)$. Thus, the problem of blind deconvolution [37] becomes the spectral estimation of the SAR system’s frequency response, $H\left(\mathbf{f}\right)$, which is necessary to build the inverse filter and recover the original whiteness of the data. Unless $H\left(\mathbf{f}\right)$ is otherwise known, its estimation can be carried out using least squares minimization in the Fourier domain using the SLC data [6]. In many cases, the tapering window is known, and its estimation is unnecessary.

With reference to an SLC image acquired by the X-band COSMO-SkyMed satellite constellation and focused with a Hamming window, Figure 2 shows the in-range power spectrum, the inverse filter used by the last block in Figure 1, and the whitened power spectrum. The in-range power spectrum shown in Figure 2b resembles a Hamming window. The power spectrum after whitening is roughly flat.

The amplitudes of original and whitened SLC images are shown in Figure 3a,b, respectively. The correlation coefficients (CCs), measured at unit lag from the complex data, are approximately equal to 0.30 in both range and azimuth before whitening and 0.05 after whitening. The estimation of image statistics from local windows of correlated samples is likely to be inaccurate because of the missing ergodicity of the set of samples. This drawback can be mitigated in part by using multiresolution analysis, where the autocorrelation function is downsampled toward the top level and survives only at the bottom layer [38].

Figure 2. Power spectra: (a) periodogram of SLC data correlated in slant-range direction; (b) frequency response of inverse filter; (c) periodogram of SLC data in (a) after whitening with the filter in (b).

Figure 2. Power spectra: (a) periodogram of SLC data correlated in slant-range direction; (b) frequency response of inverse filter; (c) periodogram of SLC data in (a) after whitening with the filter in (b).

Figure 3. Effects of SAR processor: (a) spatially correlated speckle originating from frequency windowing; (b) correlated speckle whitened using the inverse filter in Figure 2b; (c) example of a point target focused without a tapering window (negative grayscale).

Figure 3. Effects of SAR processor: (a) spatially correlated speckle originating from frequency windowing; (b) correlated speckle whitened using the inverse filter in Figure 2b; (c) example of a point target focused without a tapering window (negative grayscale).

2.1.3. Point Targets

The success of the whitening process outlined in Section 2.1.2 is diminished if speckle cannot be assumed to be fully developed. This happens wherever single-pixel reflectors or point targets are present in the scene. Point targets should be detected, removed, and substituted with patches of synthetic complex speckle, whose variance is equal to the average variance of its neighbors before the deconvolution is carried out [6]. For single-polarization images, point targets may be detected through percentile thresholding of the amplitude or intensity of the SLC image [34]. For polarimetric data, advantage may be taken of the capability of PolSAR to characterize the nature of scattering mechanisms [24].

The Fourier-domain whitening procedure [6] requires the preliminary suppression of point targets. In fact, the deconvolution process diminishes the benefits of the tapering window: the impulsive autocorrelation function is restored over distributed targets soft targets). However, on hard targets, if they are not detected, removed, replaced, and reinserted after deconvolution, the inverse filter will generate artifacts similar to the PSF of the SAR system (see Figure 3c), originating in azimuth due to the increase in the antenna sidelobes [39]. Processing point targets before the spectral whitening operation is crucial. Besides a simple yet effective thresholding, a more sophisticated detector could be used. Contextual information on targets may be exploited after detecting the “brightest” pixels because if the target is more than one pixel wide, Gibbs effects do not occur after inverse filtering. In fact, if bright spots wider than one pixel are erroneously detected, removed, and replaced before whitening and reinserted after whitening, pixels surrounding a point target exhibit high heterogeneity, which makes their speckle less developed [40,41]; thus, their spatial decorrelation is unnecessary and can be skipped.

It is noteworthy that the whitening procedure changes the relative calibration of the SLC image by a scale factor equal to the power gain of the inverse filter. The calibration is restored before point targets are inserted back into the whitened image [6].

2.2. InSAR Processing

Let $g_1(k,l)$ and $g_2(k,l)$ denote the SLC SAR data at pixel position $(k,l)$ in two co-registered observations that constitute an interferometric pair, $g=\Re \left[g\right]+j·\Im \left[g\right]$, where $\Re \left[g\right]$ and $\Im \left[g\right]$ are the zero-mean in-phase and quadrature components of the complex envelope. Let us define the interferogram as the complex cross-correlation function, namely the normalized complex covariance, between $g_1$ and $g_2$ [42]

$$C(k,l)={\displaystyle \frac{E[g_1(k,l)·g_2^{*}(k,l)]}{\sqrt{E\left[\right|g_1{(k,l)|}^2]·E[|g_2(k,l){|}^2]}}}$$

(7)

where $g^{*}$ denotes the complex conjugate of g and $E[·]$ is the expectation operator. The complex cross-correlation (7) can be written as

$$C(k,l)=\gamma (k,l)·e^{-j\varphi (k,l)}.$$

(8)

The modulus $0\le \gamma (k,l)\le 1$ is the interferometric coherence between the two images at pixel $(k,l)$. The phase $\varphi (k,l)$ is the interferometric phase, which is the relative phase between the two images. Coherence provides information about the confidence of the estimated interferometric phase and is useful for assessing the quality of interferograms. The higher the coherence, the lower the phase noise, that is, the scattering phase centers in the two observations are more similar. Thus, coherence is typically high in steady targets, such as man-made structures (buildings and infrastructure), whereas it is lower in areas characterized by rapid changes in scattering, such as vegetated areas.

Equation (7) can be estimated by replacing the expectation with the average over a local window of L neighboring pixels (L-look coherent multilooking). In this case, the phase $\varphi (k,l)$ is the maximum likelihood (ML) estimator of the interferometric phase. The extremely high noise level in SLC data makes the size of the local window, estimated using Equation (7), crucial. If the window is too large, it will degrade the spatial resolution of the estimated coherence and phase maps. If the window is too small, it will be ineffective at reducing the noise in the estimated interferometric phase [43]. In addition, small windows would overestimate coherence. To avoid this drawback, unbiased estimators, e.g., [44], can be adopted. Another source of error in the estimated coherence map depends on the spatial autocorrelation function of the speckle. Equation (7) represents the complex covariance between the two images, normalized by the product of the standard deviations (square roots of the variances) of the moduli of the individual images. While the former is insensitive to the spatial correlation because the measurements are taken across two different observations, the calculation of both variances is biased because of the spatial autocorrelation function. The underestimation of variance occurs because the local estimation domain is not ergodic in variance because the two processes are autocorrelated. A balance between spatial resolution and estimation accuracy can be achieved by using non-local (NL) estimators, which overcome the limitations of a local processing window.

Non-Local Filtering of Interferogram

The NL mean (NLM) filter relies on the concept of estimating the noise-free image as a weighted combination of surrounding noisy pixels, where the weights account for the “similarity” between the processed pixel n and its surrounding pixel m. The weights are assumed to depend on the Euclidean distance between local pixel patches centered at n and m. Analogously to Section 2.1.2, m and n denote spatial 2D coordinates.

For SAR images, weights have been tailored to the speckle statistics. The probabilistic patch-based (PPB) filter [45] extends the NLM filter to deal with SAR images as a weighted maximum likelihood estimator (WMLE). Under the WMLE principle, the noise-free image can be evaluated by maximizing a weighted likelihood function of the noisy data

$$\widehat{f}\left(n\right)=arg\underset{f}{max}\sum _{m}w(n,m)log\left[p\left(g\left(m\right)\right|f)\right]$$

(9)

where $\widehat{f}\left(n\right)$ is the estimated noise-free image $f\left(n\right)$, $g\left(n\right)$ is the observed noisy image, and the weights $w(n,m)$ can be regarded as a measure of the extent to which a pixel at m has a similar distribution to the reference pixel at n.

The formulation of the weights $w(n,m)$ is the main challenge with the WMLE. In the PPB algorithm, the solution consists of expressing the weights as the probability, conditional on the observed image g, that two pixel patches centered at positions n and m are modeled by identical distributions. If we assume that pixels in the patches are independent of one another, the weights can be formalized as

$$w(n,m)=\prod _{k}p{(f(n+k)=f(m+k)|g(n+k),g(m+k))}^{1/\alpha }$$

(10)

where k varies over the patch and $\alpha$ is a decay parameter. According to Bayesian reasoning, without knowledge of prior probabilities, $p\left(f\right(n+k)=f(m+k\left)\right)$, the posterior probabilities in Equation (10) are assumed to be proportional to the likelihood function, $p\left(g\right(n+k),g(m+k\left)\right|f(n+k)=f(m+k))$. For SAR images in intensity format, if the pixel amplitudes, $a=\sqrt{g}$, are assumed to be independent and identically distributed according to the Nakagami–Rayleigh law [34], the weights of the PPB filter can be determined as follows [45]:

$$w(n,m)=exp\left[-{\displaystyle \frac{1}{h}}\sum _{k}log\left({\displaystyle \frac{a(n+k)}{a(m+k)}}+{\displaystyle \frac{a(m+k)}{a(n+k)}}\right)\right]$$

(11)

and the speckle-free image is given by

$$\widehat{f}\left(n\right)={\displaystyle \frac{{\sum }_{m}w(n,m)a^2\left(m\right)}{{\sum }_{m}w(n,m)}}.$$

(12)

In Ref. [45], a further improvement of the model considers the probabilities in Equation (10) as also depending on previous estimates of the despeckled image, thereby leading to an iterative scheme where the weights are updated according to the previously filtered image.

The Non-Local Interferometric Estimator (NL-InSAR) [46], an extension of the PPB to InSAR data, is a state-of-the-art coherent filter that jointly yields a consistent estimate of coherence and a noise-free estimate of the interferometric phase, provided that coherence is sufficiently high.

3. Results

The experimental section is organized into two parts, each utilizing datasets from different satellites. Given an interferometric pair of SLC images in their native slant-range geometry, the two images are separately whitened and used to calculate an interferogram, from which the modulus, phase, and coherence are extracted. The goal is to demonstrate that the benefits of the whitening of SLC data are not related to a particular satellite system or a specific processing of the interferograms. In principle, the accuracy of the phase is better if the slave image of the interferometric pair is whitened after it is resampled over the master image, In practice, the two procedures are equivalent.

3.1. COSMO-SkyMed Dataset

An interferometric pair of X-band COSMO-SkyMed images, taken in StripMap mode with a pixel size of approximately 2.5 m × 2.5 m, depicting industrial buildings and vegetated areas near Pomigliano, Italy, was available. The pair was acquired by the four-satellite constellation operated by the Italian Space Agency (ASI) since 2010. The two images were processed as follows:

• Without a Hamming window, starting from raw data;
• With a Hamming window, starting from raw data;
• With a Hamming window and then spatially decorrelated.

Figure 4 displays a comparison of the incoherent multilooking of the amplitudes of the master of the interferometric pair for a closeup of the test area in the three cases. In Figure 4b, the point scatterers are better defined than those in Figure 4a, thanks to Hamming windowing. The effects of the whitening step are visible, as the speckle appears less granular in Figure 4c, while point targets are preserved.

Interferograms and related coherence and phase maps were calculated with coherent spatial multilooking for the three cases. In case 1, coherence is low in vegetation and also suffers from the spreading of areas featuring strong backscattering because of an increased number of sidelobes [47]. In case 2, point targets and buildings are better defined, thanks to the reduction in sidelobes achieved by the frequency window, but coherence is abnormally increased in the background because of the spatial correlation caused by windowing. Case 3 is the most favorable because the whitening operation produces low coherence in vegetated areas and high coherence in buildings, where the focusing benefits of the window are retained. Figure 5 shows the coherence maps in the three cases: Figure 5a,c exhibit better-defined high-coherence sections corresponding to built structures, as seen in Figure 5b, but do not suffer from anomalous increases coherence in temporally decorrelated areas (vegetation).

An examination of the phase maps shown in Figure 6 strengthens the results achieved for coherence. The phase maps show that the focusing of linear targets is better after the whitening stage. Although the phase is noisy in low-coherence areas, Figure 6c shows slightly better accuracy in high-coherence structured areas compared to the case with correlated noise in Figure 6b. The case in Figure 6a is unique: while the background speckle is uncorrelated, linear scattering structures are poorly focused and hence the accuracy of the phase map is diminished.

Here, the processing of interferograms was performed using boxcar filtering, which is the simplest tool for estimating interferometric coherence and phase maps. The goal is to objectively assess the advantages of spatial decorrelation, rather than to achieve the ultimate performance. In the next section, interferometric processing is carried out using an extremely sophisticated and powerful tool, namely NL-InSAR, which is capable of overcoming the drawbacks of local processing windows but is still affected by the spatial correlation of the speckle.

Figure 4. Interferometric pair of images of Pomigliano, master image: (a) processed without a Hamming window; (b) processed with a Hamming window; (c) processed with a Hamming window and subsequently whitened.

Figure 4. Interferometric pair of images of Pomigliano, master image: (a) processed without a Hamming window; (b) processed with a Hamming window; (c) processed with a Hamming window and subsequently whitened.

Figure 5. Coherence maps of Pomigliano estimated using 3 × 3 boxcar filtering: (a) SLC pair processed without a Hamming window; (b) SLC pair processed with a Hamming window; (c) SLC pair processed with a Hamming window and subsequently whitened.

Figure 5. Coherence maps of Pomigliano estimated using 3 × 3 boxcar filtering: (a) SLC pair processed without a Hamming window; (b) SLC pair processed with a Hamming window; (c) SLC pair processed with a Hamming window and subsequently whitened.

Figure 6. Maps of the interferometric phase of Pomigliano estimated using a 3 × 3 sliding window: (a) SLC pair processed without a Hamming window; (b) SLC pair processed with a Hamming window; (c) SLC pair processed with a Hamming window and subsequently whitened.

Figure 6. Maps of the interferometric phase of Pomigliano estimated using a 3 × 3 sliding window: (a) SLC pair processed without a Hamming window; (b) SLC pair processed with a Hamming window; (c) SLC pair processed with a Hamming window and subsequently whitened.

3.2. TerraSAR-X/TanDEM-X Dataset

The second test site consists of images collected by the X-band TerraSAR-X/TanDEM-X twin satellites (TSX-TDX) operated by DLR, the German Aerospace Center, in the area of Euskirchen, Germany. The images depict an agricultural region surrounding an urban settlement. Each image is in StripMap mode (3 m resolution). Here, temporal decorrelation is also low for vegetated areas, thanks to an extremely low temporal distance between acquisitions. The SAR processor employs a Hamming frequency window, which introduces a pronounced spatial correlation, particularly noticeable on the modulus of the unfiltered interferogram, as shown in Figure 7.

The interferograms calculated from the original and whitened SLC image pairs were processed using the NL-InSAR algorithm [46], which is a modified version of the PPB [45] and is suitable for filtering SAR interferograms. Figure 8 displays the moduli of the filtered interferograms for the original and whitened data. In the latter case, filtering is more aggressive in vegetated areas than in built-up areas, where the detail preservation accuracy of NL-InSAR is comparable to the accuracy achieved with non-whitened speckle. NL-InSAR runs with a search area of 31 × 31, 3 × 3 pixel patterns, nine looks (i.e., the nine surrounding pixels with the largest weights are weighted and averaged), and four iterations. The computational cost is significantly greater than that of the baseline 3 × 3 box filtering used in the previous simulations. Nonetheless, NL-InSAR, including its polarimetric version [48] is “de facto” recognized as a standard for the coherent processing of SAR data [49], despite its computational requirements, which can be adjusted parametrically.

The goal of interferometric processing, however, is not the smoothing of interferograms but to achieve accurate measurements of coherence, as well as spatial regularity and cleanness in the phase map. In fact, a high coherence value, usually above 0.6, indicates that the phase is reliable and can be unwrapped to reveal the topography of the scene. The coherence maps calculated by NL-InSAR from the original and whitened SLC pairs are shown in Figure 9. What immediately stands out is that the map calculated from the whitened data exhibits values 10% lower, on average, than those calculated from the non-whitened data. Interestingly, the difference is concentrated in the vegetated area and is negligible in the built-up area, as shown in Figure 10. The reason for this is the non-local processing of the interferogram. NL-InSAR finds pixels with similar surrounding patterns and averages them. In homogeneous agricultural land cover, such pixels are found close to one another and thus neighboring pixels are averaged, similar to box filtering. In spatially heterogeneous areas, statistically similar pixels are sparse and thus speckles are uncorrelated with each other. In the error map, the difference between the coherence maps calculated from whitened and non-whitened data reflects the heterogeneity of the scene: in homogeneous areas, the estimated coherence is always biased upward if the data are not whitened. In heterogeneous areas, the estimation is accurate in both cases, and the error vanishes. Unfortunately, the coherence bias is space-varying and cannot be predicted or compensated for. There is another bias effect other than correlation due to the limited sample size [42], which introduces a slight underestimation noticeable only in highly structured areas, where the bias due to correlation is negligible.

Figure 8. Modulus of interferogram of Euskirchen filtered using NL-INSAR: (a) from non-whitened SLC pair; (b) from whitened SLC pair.

Figure 8. Modulus of interferogram of Euskirchen filtered using NL-INSAR: (a) from non-whitened SLC pair; (b) from whitened SLC pair.

The maps of the interferometric phase calculated from the original and whitened data using NL-InSAR are compared in Figure 11. The high coherence of the scene results in regular and clean phase maps. The benefits of whitening, however, are noticeable in terms of a lower noise level and reduced fragmentation of the map.

A figure of merit of the quality of the phase map, in terms of its suitability for unwrapping, is provided by the residues, which are singularities in the 2D phase field that may hamper the phase-unwrapping process and compromise the accuracy of the topographic reconstruction. The residues of the phase field are overlaid on the coherence maps and shown in Figure 12 for the original and whitened data. Although the residues are mostly concentrated in heterogeneous areas, there is a 24% reduction in the number of residues for the whitened case. This supports the conclusion that the proposed whitening is beneficial not only for achieving unbiased coherence estimation but also for improving the accuracy of the phase map, regardless of the method used to process the interferogram.

Figure 9. Coherence maps of Euskirchen estimated using NL-InSAR: (a) from non-whitened SLC pair; (b) from whitened SLC pair.

Figure 9. Coherence maps of Euskirchen estimated using NL-InSAR: (a) from non-whitened SLC pair; (b) from whitened SLC pair.

Figure 10. Difference in coherence calculated from whitened and original data of Euskirchen: the overestimation due to correlation reaches 16% in homogeneous areas.

Figure 10. Difference in coherence calculated from whitened and original data of Euskirchen: the overestimation due to correlation reaches 16% in homogeneous areas.

Figure 11. Maps of interferometric phases of Euskirchen estimated using NL-InSAR: (a) from non-whitened SLC pair; (b) from whitened SLC pair.

Figure 11. Maps of interferometric phases of Euskirchen estimated using NL-InSAR: (a) from non-whitened SLC pair; (b) from whitened SLC pair.

4. Discussion

After presenting all the results, some observations can be made. The test on the COSMO-SkyMed data highlights the benefits of spatial decorrelation compared to SAR processing without a window: in the latter case, speckle is white but targets are unfocused; in the former case, speckle becomes white, and targets, which are not decorrelated, remain focused. The reason the SAR processor usually employs tapering windows in both range and azimuth is that the induced spatial correlation of speckle is considered a lesser drawback because the focusing of targets is generally mandatory.

The test on the TSX-TDX data highlights that in vegetated areas, even with extremely close acquisition passes, low coherence can still be observed due to leaves rustling in the wind. Other sources of low coherence include shadows and occlusions in built-up areas. Therefore, the accuracy of coherence measurements is important even with very close passes, which is crucial for topographic and especially tomographic applications of SAR.

While the improvement in the coherence maps is mainly linked to the spatial decorrelation of the speckle, which recovers local ergodicity and avoids underestimation of local variances in the denominator of Equation (7), the reason why the interferogram, modulus, and phase improve when calculated from whitened data is that for the same number of looks, these are uncorrelated rather than correlated. This increases the number of effective looks (i.e., with information content equal to independent ideal looks), thereby decreasing the variance of the estimated phase [50].

Figure 12. Phase residues overlaid on the coherence map of Euskirchen estimated using NL-InSAR: (a) from non-whitened SLC pair; (b) from whitened SLC pair.

Figure 12. Phase residues overlaid on the coherence map of Euskirchen estimated using NL-InSAR: (a) from non-whitened SLC pair; (b) from whitened SLC pair.

Also, in contexts where phase information is not relevant, e.g., the analysis of changes in glaciers and snow cover [16] or rainforest mapping [17], the increased accuracy of coherence analysis carried out from whitened data is advantageous.

Furthermore, in tomographic processing with whitened speckle, another effect beyond the benefits of an increased number of effective looks, similar to interferogram production, can arise. In fact, the final quality of the tomographic output may be improved by the increased range resolution intrinsically offered by the deconvolution of speckle. This could have some beneficial effects on SAR tomography implementations limited by a coarse range resolution, as seen in the upcoming P-band narrow bandwidth BIOMASS satellite system for global forest monitoring [51,52].

We wish to remark that the proposed whitening step works only on complex data. It cannot be used for incoherent multilook products. On the other hand, the downsampling operation inherent in multilooking reduces the autocorrelation function of speckle, so the whitening step is no longer necessary. Since coherent processing starts with SLC data, the relevance of whitening is guaranteed.

5. Conclusions

In this article, we have provided evidence that the spectral whitening procedure [6], originally conceived and developed as an unsupervised preprocessing method for SLC data to improve the despeckling of their modulus, may also be useful for coherent data processing. Thus, for SAR interferometry, the trade-off between the ideal situations of focused targets and uncorrelated speckle can be relaxed. Although not specifically addressed here, the unsupervised nature of the solution—namely, blind estimation and inversion of the frequency window—allows any type of correlation impairing the SLC data, including that originating from the resampling of the slave over the master, to be removed.

Possible developments will concern the investigation of the effects of whitening on multipass DInSAR processing chains, which are of key importance for applications in risk monitoring. This is especially important in mixed scenarios characterized by the presence of both rural and built-up environments and requires ad hoc preprocessing of the stack of InSAR observations [53] before selecting the set of coherent points for interferometric processing [54,55]. Future research will also test the possible benefits of applying speckle whitening in the field of SAR tomography of natural media [56].