# Comments on “PolMERLIN: Self-Supervised Polarimetric Complex SAR Image Despeckling With Masked Networks”

Loïc Denis

► To cite this version:

Loïc Denis. Comments on “PolMERLIN: Self-Supervised Polarimetric Complex SAR Image Despeckling With Masked Networks”. IEEE Geoscience and Remote Sensing Letters, 2024, 21, pp.1-2.
10.1109/LGRS.2024.3387994 . ujm-04880217

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Submitted on 10 Jan 2025

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# Comment on the paper “PolMERLIN: Self-Supervised Polarimetric Complex SAR Image Despeckling with Masked Networks”

Loïc Denis, Senior Member, IEEE

***Abstract***—A self-supervised despeckling approach based on the decomposition of single-look complex SAR images into their real and imaginary components has recently been introduced under the name MERLIN. At its core is the observation that, under Goodman's fully developed speckle model, the real and imaginary parts of single-channel SAR images are *independent and identically distributed*. The paper commented here, PolMERLIN, proposes an extension to multi-channel SAR images such as polarimetric SAR (PolSAR) images. This extension is based on the independence between real and imaginary parts of PolSAR images, but this independence generally does not hold, as shown here.

***Index Terms***—polarimetric SAR, despeckling, self-supervised learning, multivariate Gaussian models

## I. INTRODUCTION

SPECKLE is a phenomenon that occurs in synthetic aperture radar (SAR) images due to the coherent summation of the echoes back-scattered by each element located within the resolution cell. It manifests in SAR images in the form of strong fluctuations proportional to the local reflectivity. These fluctuations make the visual and automatic analysis of SAR images challenging. Many despeckling approaches have been developed through the years to improve SAR images [1], [2]. With the advent of deep learning, algorithms capable of preserving the various geometrical structures and textures found in SAR images emerged [3], [4]. In the context of despeckling, self-supervised learning is a learning strategy that only uses corrupted images. It is very attractive since it can readily include the huge amounts of SAR images captured on a daily basis by SAR satellites to learn refined models. It also avoids the biases of supervised learning that relies on pairs of speckle-free / speckled images that are available only through approximations (e.g., simulated speckle from almost speckle-free images produced by temporal or spatial averaging). MERLIN [5] proposes to separate single-look complex (SLC) images into their real and imaginary parts. Under Goodman's fully-developed speckle model, the real and imaginary parts of single-channel SAR images can be shown to be *independent and identically distributed* (i.i.d.). Hence, training a network to provide an output as close as possible to the imaginary component when only the real component is fed to the network leads to very good estimates

of the SAR reflectivity: due to the statistical independence, the stochastic component of the image can not be guessed and the deterministic component common to the real and imaginary parts is extracted (i.e., the expectation). This training strategy corresponds to an application of the noise2noise principle [6] to SAR imagery where, by essence, independent pairs of speckle realizations are captured when the complex amplitude of the back-scattered field is measured.

In [7], the Authors propose an extension of MERLIN to multi-channel SAR images, with an illustration in PolSAR. They base their extension on the statistical independence of the real and imaginary parts of diffusion vectors in SAR polarimetry. Unfortunately, as shown in section II, this independence generally does not hold. In section III, the joint distribution of the real and imaginary parts of multi-channel SAR images, such as polarimetric and / or interferometric SAR images, is derived.

## II. REAL AND IMAGINARY PARTS OF POLSAR IMAGES ARE IDENTICALLY DISTRIBUTED BUT NOT INDEPENDENT

In PolMERLIN [7], the Authors consider dual-pol SAR images where at each pixel two complex values are captured: $\zeta = [z_{hh} \ z_{vv}]^T$. Keeping the same notations as the original paper, these complex amplitudes can be decomposed in their real and imaginary parts: $z_{hh} = a_{hh} + jb_{hh}$ and $z_{vv} = a_{vv} + jb_{vv}$. As shown in [5] and recalled in section II of PolMERLIN [7], under Goodman's fully-developed speckle model, $a_{hh}$ and $b_{hh}$ are independent and identically distributed according to the Gaussian distribution<fnref n="1" /> $N(0, \frac{1}{2}r_{hh})$, and so are $a_{vv}$ and $b_{vv}$ according to $N(0, \frac{1}{2}r_{vv})$. The fact that $p(a_{hh}) = p(b_{hh})$ and $p(a_{vv}) = p(b_{vv})$ does not imply, though, that $p(a_{hh}, b_{vv}) = p(b_{hh}, a_{vv})$, as incorrectly assumed in [7] to conclude that $\text{Im}(\mathbb{E}[z_{hh}\bar{z}_{vv}]) = 0$ and, after some additional developments, that $\text{Re}(\zeta)$ and $\text{Im}(\zeta)$ are independent. In fact, $p(a_{hh}, b_{vv}) = p(b_{vv}|a_{hh})p(a_{hh}) = p(b_{vv}|a_{hh})p(b_{hh})$ (since $p(a_{hh}) = p(b_{hh})$) and $p(a_{hh}, a_{vv}) = p(b_{hh}|a_{vv})p(a_{vv}) = p(b_{hh}|a_{vv})p(b_{vv})$ (since $p(a_{vv}) = p(b_{vv})$). Therefore, if on the one hand $b_{vv}$ and $a_{hh}$ are assumed independent, and on the other hand $b_{hh}$ and $a_{vv}$ are also assumed independent, then $p(a_{hh}, b_{vv}) = p(b_{hh}, a_{vv})$ and the results of [7] hold. In the general case, though, it is not true that $\text{Re}(\zeta)$ and $\text{Im}(\zeta)$ are independent and the conclusion of section III.A in [7] is

Loïc Denis is with Université Jean Monnet Saint-Etienne, CNRS, Institut d'Optique Graduate School, Laboratoire Hubert Curien UMR 5516, F-42023, SAINT-ETIENNE, France and also with LTCI, Télécom Paris, Institut Polytechnique de Paris, Palaiseau, France, (e-mail: loic.denis@univ-st-etienne.fr).

<footnote n="1">¹since the random variables considered in this paper are jointly Gaussian, the notion of decorrelation and statistical independence can be used interchangeably</footnote>
wrong. In the next section, the joint distribution of the real and imaginary parts of multi-channel SLC SAR images is derived, in the general case of images with D channels.

### III. DERIVATION OF THE DISTRIBUTION OF THE REAL AND IMAGINARY PARTS OF MULTI-CHANNEL SAR IMAGES

Let $z \in \mathbb{C}^D$ be a D-channels SAR diffusion vector. For example, $z = [z_{hh} z_{vv}]^T \in \mathbb{C}^2$ in the dual-pol case considered in [7], $z = [z_{hh} z_{hv} z_{vv}]^T \in \mathbb{C}^3$ in mono-static full-polarimetry, $z = [z_1 z_2]^T \in \mathbb{C}^2$ in interferometry.

Let us decompose $z$ into its real and imaginary parts: $z = a + jb$, with $a$ and $b$ two vectors of $\mathbb{R}^D$. If we assume that $z \sim \mathcal{N}_c(\Sigma)$, under Goodman's fully developed speckle model, with $\mathcal{N}_c(\Sigma)$ the complex circular Gaussian distribution of covariance matrix $\Sigma$, then we can write: $z = L\epsilon$ with $L \in \mathbb{C}^{D \times D}$ a matrix such that $LL^H = \Sigma$ and $\epsilon \sim \mathcal{N}_c(I_D)$ a white speckle noise with identity covariance matrix.

Let $M = \text{Re}(L)$, $N = \text{Im}(L)$, $\mu = \text{Re}(\epsilon)$, and $\nu = \text{Im}(\epsilon)$. Since $\epsilon \sim \mathcal{N}_c(I_D)$, $\mu$ and $\nu$ are real-valued i.i.d. random vectors distributed according to the centered Gaussian distribution $\mathcal{N}(0, \frac{1}{2}I_D)$. The generative model $z = L\epsilon$ can be expanded into:

$$a + jb = (M + jN) \cdot (\mu + j\nu) \quad (1)$$

$$= M\mu - N\nu + j(N\mu + M\nu), \quad (2)$$

or, in a more compact form:

$$\begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} M & -N \\ N & M \end{bmatrix} \begin{bmatrix} \mu \\ \nu \end{bmatrix}. \quad (3)$$

Since $\begin{bmatrix} \mu \\ \nu \end{bmatrix} \sim \mathcal{N}(0, \frac{1}{2}I_{2D})$, $\begin{bmatrix} a \\ b \end{bmatrix} \sim \mathcal{N}(0, \frac{1}{2}C)$ with

$$C = \begin{bmatrix} M & -N \\ N & M \end{bmatrix} \begin{bmatrix} M^T & N^T \\ -N^T & M^T \end{bmatrix} \quad (4)$$

$$= \begin{bmatrix} MM^T + NN^T & MN^T - NM^T \\ NM^T - MN^T & MM^T + NN^T \end{bmatrix} \quad (5)$$

Given that

$$\Sigma = LL^H \quad (6)$$

$$= (M + jN)(M + jN)^H \quad (7)$$

$$= (MM^T + NN^T) + j(NM^T - MN^T) \quad (8)$$

we have

$$C = \begin{bmatrix} \text{Re}(\Sigma) & -\text{Im}(\Sigma) \\ \text{Im}(\Sigma) & \text{Re}(\Sigma) \end{bmatrix}. \quad (9)$$

Note that, since $\Sigma_{dd} \equiv \mathbb{E}[|z_d|^2]$ is real-valued for any channel $d$, the diagonal of matrix $\text{Im}(\Sigma)$ is zero and $\text{Cov}(a_d, b_d) = 0$ for any channel $d$: the real and imaginary parts of a given channel are mutually independent. Yet, although the real and imaginary parts $a$ and $b$ of the vector of complex amplitudes $z$ are identically distributed according to $\mathcal{N}(0, \frac{1}{2}\text{Re}(\Sigma))$, they are generally not jointly independent. Their covariance $\text{Cov}(a, b) = -\text{Im}(\Sigma)$ is non-zero unless $\Sigma$ is either diagonal (indicating zero coherence) or $\Sigma$ is real-valued (corresponding to 0 or $\pi$ phase shifts between the different channels).

As a consequence, MERLIN's self-supervised learning framework cannot be directly applied to polarimetric or interferometric SAR images.

### IV. CONCLUSION

The aim of this comment was to correct the statement that real and imaginary parts of multi-channel SAR images are statistically independent. This is generally not the case, which means that the mathematical conditions to apply the self-supervised training strategy of MERLIN are not verified for those images. In practice, if the coherence is low, multi-channel processing is beneficial and can outweigh the downsides of applying self-supervised learning under (weak) statistical dependencies. Experiments conducted in multi-temporal despeckling (see Fig. 6 in [8]) indicate that coherence values between 0 and 0.2 induce very limited performance degradation. The results obtained in dual polarimetry in [7] also support the application of self-supervised learning in these low-coherence configurations. The addition of the spatial masking proposed in [7] probably helps to reduce the impact of correlations between real and imaginary channels. Yet, in most cases and, in particular, in interferometric SAR or full-polarimetry, other self-supervised learning strategies remain to be imagined.

### ACKNOWLEDGMENT

The author would like to thank the French National Research Agency (ANR) and the Direction Générale de l'Armement (DGA) under ASTRAL project ANR-21-ASTR-0011 for funding this research. He is also grateful to Florence Tupin and Emanuele Dalsasso for their feedback and to the anonymous Reviewers for their comments and suggestions.

### REFERENCES

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[2] C.-A. Deledalle, L. Denis, G. Poggi, F. Tupin, and L. Verdoliva, "Exploiting patch similarity for SAR image processing: The nonlocal paradigm," *IEEE Signal Processing Magazine*, vol. 31, no. 4, pp. 69–78, 2014.

[3] B. Rasti, Y. Chang, E. Dalsasso, L. Denis, and P. Ghamisi, "Image restoration for remote sensing: Overview and toolbox," *IEEE Geoscience and Remote Sensing Magazine*, vol. 10, no. 2, pp. 201–230, 2021.

[4] G. Fracastoro, E. Magli, G. Poggi, G. Scarpa, D. Valsesia, and L. Verdoliva, "Deep learning methods for synthetic aperture radar image despeckling: An overview of trends and perspectives," *IEEE Geoscience and Remote Sensing Magazine*, vol. 9, no. 2, pp. 29–51, 2021.

[5] E. Dalsasso, L. Denis, and F. Tupin, "As if by magic: self-supervised training of deep despeckling networks with merlin," *IEEE Transactions on Geoscience and Remote Sensing*, vol. 60, pp. 1–13, 2021.

[6] J. Lehtinen, J. Munkberg, J. Hasselgren, S. Laine, T. Karras, M. Aittala, and T. Aila, "Noise2noise: Learning image restoration without clean data," *arXiv preprint arXiv:1803.04189*, 2018.

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